ILASLC scans A for its last non-zero column.

M

          M is INTEGER


          The number of rows of the matrix A.

N

          N is INTEGER


          The number of columns of the matrix A.

A

          A is REAL array, dimension (LDA,N)


          The m by n matrix A.

LDA

          LDA is INTEGER


          The leading dimension of the array A. LDA >= max(1,M).

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 ILASLR scans A for its last non-zero row.

M

          M is INTEGER


          The number of rows of the matrix A.

N

          N is INTEGER


          The number of columns of the matrix A.

A

          A is REAL array, dimension (LDA,N)


          The m by n matrix A.

LDA

          LDA is INTEGER


          The leading dimension of the array A. LDA >= max(1,M).

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 SLABRD reduces the first NB rows and columns of a real general


 m by n matrix A to upper or lower bidiagonal form by an orthogonal


 transformation Q**T * A * P, and returns the matrices X and Y which


 are needed to apply the transformation to the unreduced part of A.


 If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower


 bidiagonal form.


 This is an auxiliary routine called by SGEBRD

M

          M is INTEGER


          The number of rows in the matrix A.

N

          N is INTEGER


          The number of columns in the matrix A.

NB

          NB is INTEGER


          The number of leading rows and columns of A to be reduced.

A

          A is REAL array, dimension (LDA,N)


          On entry, the m by n general matrix to be reduced.


          On exit, the first NB rows and columns of the matrix are


          overwritten; the rest of the array is unchanged.


          If m >= n, elements on and below the diagonal in the first NB


            columns, with the array TAUQ, represent the orthogonal


            matrix Q as a product of elementary reflectors; and


            elements above the diagonal in the first NB rows, with the


            array TAUP, represent the orthogonal matrix P as a product


            of elementary reflectors.


          If m < n, elements below the diagonal in the first NB


            columns, with the array TAUQ, represent the orthogonal


            matrix Q as a product of elementary reflectors, and


            elements on and above the diagonal in the first NB rows,


            with the array TAUP, represent the orthogonal matrix P as


            a product of elementary reflectors.


          See Further Details.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is REAL array, dimension (NB)


          The diagonal elements of the first NB rows and columns of


          the reduced matrix.  D(i) = A(i,i).

E

          E is REAL array, dimension (NB)


          The off-diagonal elements of the first NB rows and columns of


          the reduced matrix.

TAUQ

          TAUQ is REAL array, dimension (NB)


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix Q. See Further Details.

TAUP

          TAUP is REAL array, dimension (NB)


          The scalar factors of the elementary reflectors which


          represent the orthogonal matrix P. See Further Details.

X

          X is REAL array, dimension (LDX,NB)


          The m-by-nb matrix X required to update the unreduced part


          of A.

LDX

          LDX is INTEGER


          The leading dimension of the array X. LDX >= max(1,M).

Y

          Y is REAL array, dimension (LDY,NB)


          The n-by-nb matrix Y required to update the unreduced part


          of A.

LDY

          LDY is INTEGER


          The leading dimension of the array Y. LDY >= max(1,N).

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  The matrices Q and P are represented as products of elementary


  reflectors:


     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)


  Each H(i) and G(i) has the form:


     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T


  where tauq and taup are real scalars, and v and u are real vectors.


  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in


  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in


  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).


  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in


  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in


  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).


  The elements of the vectors v and u together form the m-by-nb matrix


  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply


  the transformation to the unreduced part of the matrix, using a block


  update of the form:  A := A - V*Y**T - X*U**T.


  The contents of A on exit are illustrated by the following examples


  with nb = 2:


  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):


    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )


    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )


    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )


    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )


    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )


    (  v1  v2  a   a   a  )


  where a denotes an element of the original matrix which is unchanged,


  vi denotes an element of the vector defining H(i), and ui an element


  of the vector defining G(i).

 SLACN2 estimates the 1-norm of a square, real matrix A.


 Reverse communication is used for evaluating matrix-vector products.

N

          N is INTEGER


         The order of the matrix.  N >= 1.

V

          V is REAL array, dimension (N)


         On the final return, V = A*W,  where  EST = norm(V)/norm(W)


         (W is not returned).

X

          X is REAL array, dimension (N)


         On an intermediate return, X should be overwritten by


               A * X,   if KASE=1,


               A**T * X,  if KASE=2,


         and SLACN2 must be re-called with all the other parameters


         unchanged.

ISGN

          ISGN is INTEGER array, dimension (N)

EST

          EST is REAL


         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be


         unchanged from the previous call to SLACN2.


         On exit, EST is an estimate (a lower bound) for norm(A).

KASE

          KASE is INTEGER


         On the initial call to SLACN2, KASE should be 0.


         On an intermediate return, KASE will be 1 or 2, indicating


         whether X should be overwritten by A * X  or A**T * X.


         On the final return from SLACN2, KASE will again be 0.

ISAVE

          ISAVE is INTEGER array, dimension (3)


         ISAVE is used to save variables between calls to SLACN2

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  Originally named SONEST, dated March 16, 1988.


  This is a thread safe version of SLACON, which uses the array ISAVE


  in place of a SAVE statement, as follows:


     SLACON     SLACN2


      JUMP     ISAVE(1)


      J        ISAVE(2)


      ITER     ISAVE(3)

Nick Higham, University of Manchester

N.J. Higham, 'FORTRAN codes for estimating the one-norm of
a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

 SLACON estimates the 1-norm of a square, real matrix A.


 Reverse communication is used for evaluating matrix-vector products.

N

          N is INTEGER


         The order of the matrix.  N >= 1.

V

          V is REAL array, dimension (N)


         On the final return, V = A*W,  where  EST = norm(V)/norm(W)


         (W is not returned).

X

          X is REAL array, dimension (N)


         On an intermediate return, X should be overwritten by


               A * X,   if KASE=1,


               A**T * X,  if KASE=2,


         and SLACON must be re-called with all the other parameters


         unchanged.

ISGN

          ISGN is INTEGER array, dimension (N)

EST

          EST is REAL


         On entry with KASE = 1 or 2 and JUMP = 3, EST should be


         unchanged from the previous call to SLACON.


         On exit, EST is an estimate (a lower bound) for norm(A).

KASE

          KASE is INTEGER


         On the initial call to SLACON, KASE should be 0.


         On an intermediate return, KASE will be 1 or 2, indicating


         whether X should be overwritten by A * X  or A**T * X.


         On the final return from SLACON, KASE will again be 0.

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Nick Higham, University of Manchester.
Originally named SONEST, dated March 16, 1988.

N.J. Higham, 'FORTRAN codes for estimating the one-norm of
a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

 SLADIV performs complex division in  real arithmetic


                       a + i*b


            p + i*q = ---------


                       c + i*d


 The algorithm is due to Michael Baudin and Robert L. Smith


 and can be found in the paper


 'A Robust Complex Division in Scilab'

A

          A is REAL

B

          B is REAL

C

          C is REAL

D

          D is REAL


          The scalars a, b, c, and d in the above expression.

P

          P is REAL

Q

          Q is REAL


          The scalars p and q in the above expression.

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 SLAEIN uses inverse iteration to find a right or left eigenvector


 corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg


 matrix H.

RIGHTV

          RIGHTV is LOGICAL


          = .TRUE. : compute right eigenvector;


          = .FALSE.: compute left eigenvector.

NOINIT

          NOINIT is LOGICAL


          = .TRUE. : no initial vector supplied in (VR,VI).


          = .FALSE.: initial vector supplied in (VR,VI).

N

          N is INTEGER


          The order of the matrix H.  N >= 0.

H

          H is REAL array, dimension (LDH,N)


          The upper Hessenberg matrix H.

LDH

          LDH is INTEGER


          The leading dimension of the array H.  LDH >= max(1,N).

WR

          WR is REAL

WI

          WI is REAL


          The real and imaginary parts of the eigenvalue of H whose


          corresponding right or left eigenvector is to be computed.

VR

          VR is REAL array, dimension (N)

VI

          VI is REAL array, dimension (N)


          On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain


          a real starting vector for inverse iteration using the real


          eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI


          must contain the real and imaginary parts of a complex


          starting vector for inverse iteration using the complex


          eigenvalue (WR,WI); otherwise VR and VI need not be set.


          On exit, if WI = 0.0 (real eigenvalue), VR contains the


          computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),


          VR and VI contain the real and imaginary parts of the


          computed complex eigenvector. The eigenvector is normalized


          so that the component of largest magnitude has magnitude 1;


          here the magnitude of a complex number (x,y) is taken to be


          |x| + |y|.


          VI is not referenced if WI = 0.0.

B

          B is REAL array, dimension (LDB,N)

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= N+1.

WORK

          WORK is REAL array, dimension (N)

EPS3

          EPS3 is REAL


          A small machine-dependent value which is used to perturb


          close eigenvalues, and to replace zero pivots.

SMLNUM

          SMLNUM is REAL


          A machine-dependent value close to the underflow threshold.

BIGNUM

          BIGNUM is REAL


          A machine-dependent value close to the overflow threshold.

INFO

          INFO is INTEGER


          = 0:  successful exit


          = 1:  inverse iteration did not converge; VR is set to the


                last iterate, and so is VI if WI.ne.0.0.

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 SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in


 an upper quasi-triangular matrix T by an orthogonal similarity


 transformation.


 T must be in Schur canonical form, that is, block upper triangular


 with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block


 has its diagonal elements equal and its off-diagonal elements of


 opposite sign.

WANTQ

          WANTQ is LOGICAL


          = .TRUE. : accumulate the transformation in the matrix Q;


          = .FALSE.: do not accumulate the transformation.

N

          N is INTEGER


          The order of the matrix T. N >= 0.

T

          T is REAL array, dimension (LDT,N)


          On entry, the upper quasi-triangular matrix T, in Schur


          canonical form.


          On exit, the updated matrix T, again in Schur canonical form.

LDT

          LDT is INTEGER


          The leading dimension of the array T. LDT >= max(1,N).

Q

          Q is REAL array, dimension (LDQ,N)


          On entry, if WANTQ is .TRUE., the orthogonal matrix Q.


          On exit, if WANTQ is .TRUE., the updated matrix Q.


          If WANTQ is .FALSE., Q is not referenced.

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q.


          LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.

J1

          J1 is INTEGER


          The index of the first row of the first block T11.

N1

          N1 is INTEGER


          The order of the first block T11. N1 = 0, 1 or 2.

N2

          N2 is INTEGER


          The order of the second block T22. N2 = 0, 1 or 2.

WORK

          WORK is REAL array, dimension (N)

INFO

          INFO is INTEGER


          = 0: successful exit


          = 1: the transformed matrix T would be too far from Schur


               form; the blocks are not swapped and T and Q are


               unchanged.

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 SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue


 problem  A - w B, with scaling as necessary to avoid over-/underflow.


 The scaling factor 's' results in a modified eigenvalue equation


     s A - w B


 where  s  is a non-negative scaling factor chosen so that  w,  w B,


 and  s A  do not overflow and, if possible, do not underflow, either.

A

          A is REAL array, dimension (LDA, 2)


          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm


          is less than 1/SAFMIN.  Entries less than


          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= 2.

B

          B is REAL array, dimension (LDB, 2)


          On entry, the 2 x 2 upper triangular matrix B.  It is


          assumed that the one-norm of B is less than 1/SAFMIN.  The


          diagonals should be at least sqrt(SAFMIN) times the largest


          element of B (in absolute value); if a diagonal is smaller


          than that, then  +/- sqrt(SAFMIN) will be used instead of


          that diagonal.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= 2.

SAFMIN

          SAFMIN is REAL


          The smallest positive number s.t. 1/SAFMIN does not


          overflow.  (This should always be SLAMCH('S') -- it is an


          argument in order to avoid having to call SLAMCH frequently.)

SCALE1

          SCALE1 is REAL


          A scaling factor used to avoid over-/underflow in the


          eigenvalue equation which defines the first eigenvalue.  If


          the eigenvalues are complex, then the eigenvalues are


          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the


          exponent range of the machine), SCALE1=SCALE2, and SCALE1


          will always be positive.  If the eigenvalues are real, then


          the first (real) eigenvalue is  WR1 / SCALE1 , but this may


          overflow or underflow, and in fact, SCALE1 may be zero or


          less than the underflow threshold if the exact eigenvalue


          is sufficiently large.

SCALE2

          SCALE2 is REAL


          A scaling factor used to avoid over-/underflow in the


          eigenvalue equation which defines the second eigenvalue.  If


          the eigenvalues are complex, then SCALE2=SCALE1.  If the


          eigenvalues are real, then the second (real) eigenvalue is


          WR2 / SCALE2 , but this may overflow or underflow, and in


          fact, SCALE2 may be zero or less than the underflow


          threshold if the exact eigenvalue is sufficiently large.

WR1

          WR1 is REAL


          If the eigenvalue is real, then WR1 is SCALE1 times the


          eigenvalue closest to the (2,2) element of A B**(-1).  If the


          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real


          part of the eigenvalues.

WR2

          WR2 is REAL


          If the eigenvalue is real, then WR2 is SCALE2 times the


          other eigenvalue.  If the eigenvalue is complex, then


          WR1=WR2 is SCALE1 times the real part of the eigenvalues.

WI

          WI is REAL


          If the eigenvalue is real, then WI is zero.  If the


          eigenvalue is complex, then WI is SCALE1 times the imaginary


          part of the eigenvalues.  WI will always be non-negative.

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 SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such


 that if ( UPPER ) then


           U**T *A*Q = U**T *( A1 A2 )*Q = ( x  0  )


                             ( 0  A3 )     ( x  x  )


 and


           V**T*B*Q = V**T *( B1 B2 )*Q = ( x  0  )


                            ( 0  B3 )     ( x  x  )


 or if ( .NOT.UPPER ) then


           U**T *A*Q = U**T *( A1 0  )*Q = ( x  x  )


                             ( A2 A3 )     ( 0  x  )


 and


           V**T*B*Q = V**T*( B1 0  )*Q = ( x  x  )


                           ( B2 B3 )     ( 0  x  )


 The rows of the transformed A and B are parallel, where


   U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )


       ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )


 Z**T denotes the transpose of Z.

UPPER

          UPPER is LOGICAL


          = .TRUE.: the input matrices A and B are upper triangular.


          = .FALSE.: the input matrices A and B are lower triangular.

A1

          A1 is REAL

A2

          A2 is REAL

A3

          A3 is REAL


          On entry, A1, A2 and A3 are elements of the input 2-by-2


          upper (lower) triangular matrix A.

B1

          B1 is REAL

B2

          B2 is REAL

B3

          B3 is REAL


          On entry, B1, B2 and B3 are elements of the input 2-by-2


          upper (lower) triangular matrix B.

CSU

          CSU is REAL

SNU

          SNU is REAL


          The desired orthogonal matrix U.

CSV

          CSV is REAL

SNV

          SNV is REAL


          The desired orthogonal matrix V.

CSQ

          CSQ is REAL

SNQ

          SNQ is REAL


          The desired orthogonal matrix Q.

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 SLAGTM performs a matrix-vector product of the form


    B := alpha * A * X + beta * B


 where A is a tridiagonal matrix of order N, B and X are N by NRHS


 matrices, and alpha and beta are real scalars, each of which may be


 0., 1., or -1.

TRANS

          TRANS is CHARACTER*1


          Specifies the operation applied to A.


          = 'N':  No transpose, B := alpha * A * X + beta * B


          = 'T':  Transpose,    B := alpha * A'* X + beta * B


          = 'C':  Conjugate transpose = Transpose

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices X and B.

ALPHA

          ALPHA is REAL


          The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,


          it is assumed to be 0.

DL

          DL is REAL array, dimension (N-1)


          The (n-1) sub-diagonal elements of T.

D

          D is REAL array, dimension (N)


          The diagonal elements of T.

DU

          DU is REAL array, dimension (N-1)


          The (n-1) super-diagonal elements of T.

X

          X is REAL array, dimension (LDX,NRHS)


          The N by NRHS matrix X.

LDX

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(N,1).

BETA

          BETA is REAL


          The scalar beta.  BETA must be 0., 1., or -1.; otherwise,


          it is assumed to be 1.

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the N by NRHS matrix B.


          On exit, B is overwritten by the matrix expression


          B := alpha * A * X + beta * B.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(N,1).

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 SLAGV2 computes the Generalized Schur factorization of a real 2-by-2


 matrix pencil (A,B) where B is upper triangular. This routine


 computes orthogonal (rotation) matrices given by CSL, SNL and CSR,


 SNR such that


 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0


    types), then


    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]


    [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]


    [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]


    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],


 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,


    then


    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]


    [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]


    [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]


    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]


    where b11 >= b22 > 0.

A

          A is REAL array, dimension (LDA, 2)


          On entry, the 2 x 2 matrix A.


          On exit, A is overwritten by the ``A-part'' of the


          generalized Schur form.

LDA

          LDA is INTEGER


          THe leading dimension of the array A.  LDA >= 2.

B

          B is REAL array, dimension (LDB, 2)


          On entry, the upper triangular 2 x 2 matrix B.


          On exit, B is overwritten by the ``B-part'' of the


          generalized Schur form.

LDB

          LDB is INTEGER


          THe leading dimension of the array B.  LDB >= 2.

ALPHAR

          ALPHAR is REAL array, dimension (2)

ALPHAI

          ALPHAI is REAL array, dimension (2)

BETA

          BETA is REAL array, dimension (2)


          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the


          pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may


          be zero.

CSL

          CSL is REAL


          The cosine of the left rotation matrix.

SNL

          SNL is REAL


          The sine of the left rotation matrix.

CSR

          CSR is REAL


          The cosine of the right rotation matrix.

SNR

          SNR is REAL


          The sine of the right rotation matrix.

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Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

    SLAHQR is an auxiliary routine called by SHSEQR to update the


    eigenvalues and Schur decomposition already computed by SHSEQR, by


    dealing with the Hessenberg submatrix in rows and columns ILO to


    IHI.

WANTT

          WANTT is LOGICAL


          = .TRUE. : the full Schur form T is required;


          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL


          = .TRUE. : the matrix of Schur vectors Z is required;


          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER


          The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          It is assumed that H is already upper quasi-triangular in


          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless


          ILO = 1). SLAHQR works primarily with the Hessenberg


          submatrix in rows and columns ILO to IHI, but applies


          transformations to all of H if WANTT is .TRUE..


          1 <= ILO <= max(1,IHI); IHI <= N.

H

          H is REAL array, dimension (LDH,N)


          On entry, the upper Hessenberg matrix H.


          On exit, if INFO is zero and if WANTT is .TRUE., H is upper


          quasi-triangular in rows and columns ILO:IHI, with any


          2-by-2 diagonal blocks in standard form. If INFO is zero


          and WANTT is .FALSE., the contents of H are unspecified on


          exit.  The output state of H if INFO is nonzero is given


          below under the description of INFO.

LDH

          LDH is INTEGER


          The leading dimension of the array H. LDH >= max(1,N).

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)


          The real and imaginary parts, respectively, of the computed


          eigenvalues ILO to IHI are stored in the corresponding


          elements of WR and WI. If two eigenvalues are computed as a


          complex conjugate pair, they are stored in consecutive


          elements of WR and WI, say the i-th and (i+1)th, with


          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the


          eigenvalues are stored in the same order as on the diagonal


          of the Schur form returned in H, with WR(i) = H(i,i), and, if


          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,


          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


          Specify the rows of Z to which transformations must be


          applied if WANTZ is .TRUE..


          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is REAL array, dimension (LDZ,N)


          If WANTZ is .TRUE., on entry Z must contain the current


          matrix Z of transformations accumulated by SHSEQR, and on


          exit Z has been updated; transformations are applied only to


          the submatrix Z(ILOZ:IHIZ,ILO:IHI).


          If WANTZ is .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= max(1,N).

INFO

          INFO is INTEGER


           = 0:   successful exit


           > 0:   If INFO = i, SLAHQR failed to compute all the


                  eigenvalues ILO to IHI in a total of 30 iterations


                  per eigenvalue; elements i+1:ihi of WR and WI


                  contain those eigenvalues which have been


                  successfully computed.


                  If INFO > 0 and WANTT is .FALSE., then on exit,


                  the remaining unconverged eigenvalues are the


                  eigenvalues of the upper Hessenberg matrix rows


                  and columns ILO through INFO of the final, output


                  value of H.


                  If INFO > 0 and WANTT is .TRUE., then on exit


          (*)       (initial value of H)*U  = U*(final value of H)


                  where U is an orthogonal matrix.    The final


                  value of H is upper Hessenberg and triangular in


                  rows and columns INFO+1 through IHI.


                  If INFO > 0 and WANTZ is .TRUE., then on exit


                      (final value of Z)  = (initial value of Z)*U


                  where U is the orthogonal matrix in (*)


                  (regardless of the value of WANTT.)

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     02-96 Based on modifications by


     David Day, Sandia National Laboratory, USA


     12-04 Further modifications by


     Ralph Byers, University of Kansas, USA


     This is a modified version of SLAHQR from LAPACK version 3.0.


     It is (1) more robust against overflow and underflow and


     (2) adopts the more conservative Ahues & Tisseur stopping


     criterion (LAWN 122, 1997).

 SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)


 matrix A so that elements below the k-th subdiagonal are zero. The


 reduction is performed by an orthogonal similarity transformation


 Q**T * A * Q. The routine returns the matrices V and T which determine


 Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.


 This is an auxiliary routine called by SGEHRD.

N

          N is INTEGER


          The order of the matrix A.

K

          K is INTEGER


          The offset for the reduction. Elements below the k-th


          subdiagonal in the first NB columns are reduced to zero.


          K < N.

NB

          NB is INTEGER


          The number of columns to be reduced.

A

          A is REAL array, dimension (LDA,N-K+1)


          On entry, the n-by-(n-k+1) general matrix A.


          On exit, the elements on and above the k-th subdiagonal in


          the first NB columns are overwritten with the corresponding


          elements of the reduced matrix; the elements below the k-th


          subdiagonal, with the array TAU, represent the matrix Q as a


          product of elementary reflectors. The other columns of A are


          unchanged. See Further Details.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

TAU

          TAU is REAL array, dimension (NB)


          The scalar factors of the elementary reflectors. See Further


          Details.

T

          T is REAL array, dimension (LDT,NB)


          The upper triangular matrix T.

LDT

          LDT is INTEGER


          The leading dimension of the array T.  LDT >= NB.

Y

          Y is REAL array, dimension (LDY,NB)


          The n-by-nb matrix Y.

LDY

          LDY is INTEGER


          The leading dimension of the array Y. LDY >= N.

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  The matrix Q is represented as a product of nb elementary reflectors


     Q = H(1) H(2) . . . H(nb).


  Each H(i) has the form


     H(i) = I - tau * v * v**T


  where tau is a real scalar, and v is a real vector with


  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in


  A(i+k+1:n,i), and tau in TAU(i).


  The elements of the vectors v together form the (n-k+1)-by-nb matrix


  V which is needed, with T and Y, to apply the transformation to the


  unreduced part of the matrix, using an update of the form:


  A := (I - V*T*V**T) * (A - Y*V**T).


  The contents of A on exit are illustrated by the following example


  with n = 7, k = 3 and nb = 2:


     ( a   a   a   a   a )


     ( a   a   a   a   a )


     ( a   a   a   a   a )


     ( h   h   a   a   a )


     ( v1  h   a   a   a )


     ( v1  v2  a   a   a )


     ( v1  v2  a   a   a )


  where a denotes an element of the original matrix A, h denotes a


  modified element of the upper Hessenberg matrix H, and vi denotes an


  element of the vector defining H(i).


  This subroutine is a slight modification of LAPACK-3.0's SLAHRD


  incorporating improvements proposed by Quintana-Orti and Van de


  Gejin. Note that the entries of A(1:K,2:NB) differ from those


  returned by the original LAPACK-3.0's SLAHRD routine. (This


  subroutine is not backward compatible with LAPACK-3.0's SLAHRD.)

Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

 SLAIC1 applies one step of incremental condition estimation in


 its simplest version:


 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j


 lower triangular matrix L, such that


          twonorm(L*x) = sest


 Then SLAIC1 computes sestpr, s, c such that


 the vector


                 [ s*x ]


          xhat = [  c  ]


 is an approximate singular vector of


                 [ L      0  ]


          Lhat = [ w**T gamma ]


 in the sense that


          twonorm(Lhat*xhat) = sestpr.


 Depending on JOB, an estimate for the largest or smallest singular


 value is computed.


 Note that [s c]**T and sestpr**2 is an eigenpair of the system


     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]


                                           [ gamma ]


 where  alpha =  x**T*w.

JOB

          JOB is INTEGER


          = 1: an estimate for the largest singular value is computed.


          = 2: an estimate for the smallest singular value is computed.

J

          J is INTEGER


          Length of X and W

X

          X is REAL array, dimension (J)


          The j-vector x.

SEST

          SEST is REAL


          Estimated singular value of j by j matrix L

W

          W is REAL array, dimension (J)


          The j-vector w.

GAMMA

          GAMMA is REAL


          The diagonal element gamma.

SESTPR

          SESTPR is REAL


          Estimated singular value of (j+1) by (j+1) matrix Lhat.

S

          S is REAL


          Sine needed in forming xhat.

C

          C is REAL


          Cosine needed in forming xhat.

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 SLALN2 solves a system of the form  (ca A - w D ) X = s B


 or (ca A**T - w D) X = s B   with possible scaling ('s') and


 perturbation of A.  (A**T means A-transpose.)


 A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA


 real diagonal matrix, w is a real or complex value, and X and B are


 NA x 1 matrices -- real if w is real, complex if w is complex.  NA


 may be 1 or 2.


 If w is complex, X and B are represented as NA x 2 matrices,


 the first column of each being the real part and the second


 being the imaginary part.


 's' is a scaling factor (<= 1), computed by SLALN2, which is


 so chosen that X can be computed without overflow.  X is further


 scaled if necessary to assure that norm(ca A - w D)*norm(X) is less


 than overflow.


 If both singular values of (ca A - w D) are less than SMIN,


 SMIN*identity will be used instead of (ca A - w D).  If only one


 singular value is less than SMIN, one element of (ca A - w D) will be


 perturbed enough to make the smallest singular value roughly SMIN.


 If both singular values are at least SMIN, (ca A - w D) will not be


 perturbed.  In any case, the perturbation will be at most some small


 multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values


 are computed by infinity-norm approximations, and thus will only be


 correct to a factor of 2 or so.


 Note: all input quantities are assumed to be smaller than overflow


 by a reasonable factor.  (See BIGNUM.)

LTRANS

          LTRANS is LOGICAL


          =.TRUE.:  A-transpose will be used.


          =.FALSE.: A will be used (not transposed.)

NA

          NA is INTEGER


          The size of the matrix A.  It may (only) be 1 or 2.

NW

          NW is INTEGER


          1 if 'w' is real, 2 if 'w' is complex.  It may only be 1


          or 2.

SMIN

          SMIN is REAL


          The desired lower bound on the singular values of A.  This


          should be a safe distance away from underflow or overflow,


          say, between (underflow/machine precision) and  (machine


          precision * overflow ).  (See BIGNUM and ULP.)

CA

          CA is REAL


          The coefficient c, which A is multiplied by.

A

          A is REAL array, dimension (LDA,NA)


          The NA x NA matrix A.

LDA

          LDA is INTEGER


          The leading dimension of A.  It must be at least NA.

D1

          D1 is REAL


          The 1,1 element in the diagonal matrix D.

D2

          D2 is REAL


          The 2,2 element in the diagonal matrix D.  Not used if NA=1.

B

          B is REAL array, dimension (LDB,NW)


          The NA x NW matrix B (right-hand side).  If NW=2 ('w' is


          complex), column 1 contains the real part of B and column 2


          contains the imaginary part.

LDB

          LDB is INTEGER


          The leading dimension of B.  It must be at least NA.

WR

          WR is REAL


          The real part of the scalar 'w'.

WI

          WI is REAL


          The imaginary part of the scalar 'w'.  Not used if NW=1.

X

          X is REAL array, dimension (LDX,NW)


          The NA x NW matrix X (unknowns), as computed by SLALN2.


          If NW=2 ('w' is complex), on exit, column 1 will contain


          the real part of X and column 2 will contain the imaginary


          part.

LDX

          LDX is INTEGER


          The leading dimension of X.  It must be at least NA.

SCALE

          SCALE is REAL


          The scale factor that B must be multiplied by to insure


          that overflow does not occur when computing X.  Thus,


          (ca A - w D) X  will be SCALE*B, not B (ignoring


          perturbations of A.)  It will be at most 1.

XNORM

          XNORM is REAL


          The infinity-norm of X, when X is regarded as an NA x NW


          real matrix.

INFO

          INFO is INTEGER


          An error flag.  It will be set to zero if no error occurs,


          a negative number if an argument is in error, or a positive


          number if  ca A - w D  had to be perturbed.


          The possible values are:


          = 0: No error occurred, and (ca A - w D) did not have to be


                 perturbed.


          = 1: (ca A - w D) had to be perturbed to make its smallest


               (or only) singular value greater than SMIN.


          NOTE: In the interests of speed, this routine does not


                check the inputs for errors.

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 SLANGT  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 real tridiagonal matrix A.

SLANGT

    SLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANGT as described


          above.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.  When N = 0, SLANGT is


          set to zero.

DL

          DL is REAL array, dimension (N-1)


          The (n-1) sub-diagonal elements of A.

D

          D is REAL array, dimension (N)


          The diagonal elements of A.

DU

          DU is REAL array, dimension (N-1)


          The (n-1) super-diagonal elements of A.

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 SLANHS  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 Hessenberg matrix A.

SLANHS

    SLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANHS as described


          above.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.  When N = 0, SLANHS is


          set to zero.

A

          A is REAL array, dimension (LDA,N)


          The n by n upper Hessenberg matrix A; the part of A below the


          first sub-diagonal is not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(N,1).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= N when NORM = 'I'; otherwise, WORK is not


          referenced.

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 SLANSB  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the element of  largest absolute value  of an


 n by n symmetric band matrix A,  with k super-diagonals.

SLANSB

    SLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANSB as described


          above.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          band matrix A is supplied.


          = 'U':  Upper triangular part is supplied


          = 'L':  Lower triangular part is supplied

N

          N is INTEGER


          The order of the matrix A.  N >= 0.  When N = 0, SLANSB is


          set to zero.

K

          K is INTEGER


          The number of super-diagonals or sub-diagonals of the


          band matrix A.  K >= 0.

AB

          AB is REAL array, dimension (LDAB,N)


          The upper or lower triangle of the symmetric band matrix A,


          stored in the first K+1 rows of AB.  The j-th column of A is


          stored in the j-th column of the array AB as follows:


          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;


          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

LDAB

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= K+1.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,


          WORK is not referenced.

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 SLANSP  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 real symmetric matrix A,  supplied in packed form.

SLANSP

    SLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANSP as described


          above.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          symmetric matrix A is supplied.


          = 'U':  Upper triangular part of A is supplied


          = 'L':  Lower triangular part of A is supplied

N

          N is INTEGER


          The order of the matrix A.  N >= 0.  When N = 0, SLANSP is


          set to zero.

AP

          AP is REAL array, dimension (N*(N+1)/2)


          The upper or lower triangle of the symmetric matrix A, packed


          columnwise in a linear array.  The j-th column of A is stored


          in the array AP as follows:


          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;


          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,


          WORK is not referenced.

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 SLANTB  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the element of  largest absolute value  of an


 n by n triangular band matrix A,  with ( k + 1 ) diagonals.

SLANTB

    SLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANTB as described


          above.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the matrix A is upper or lower triangular.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

DIAG

          DIAG is CHARACTER*1


          Specifies whether or not the matrix A is unit triangular.


          = 'N':  Non-unit triangular


          = 'U':  Unit triangular

N

          N is INTEGER


          The order of the matrix A.  N >= 0.  When N = 0, SLANTB is


          set to zero.

K

          K is INTEGER


          The number of super-diagonals of the matrix A if UPLO = 'U',


          or the number of sub-diagonals of the matrix A if UPLO = 'L'.


          K >= 0.

AB

          AB is REAL array, dimension (LDAB,N)


          The upper or lower triangular band matrix A, stored in the


          first k+1 rows of AB.  The j-th column of A is stored


          in the j-th column of the array AB as follows:


          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;


          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).


          Note that when DIAG = 'U', the elements of the array AB


          corresponding to the diagonal elements of the matrix A are


          not referenced, but are assumed to be one.

LDAB

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= K+1.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= N when NORM = 'I'; otherwise, WORK is not


          referenced.

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 SLANTP  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 triangular matrix A, supplied in packed form.

SLANTP

    SLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANTP as described


          above.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the matrix A is upper or lower triangular.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

DIAG

          DIAG is CHARACTER*1


          Specifies whether or not the matrix A is unit triangular.


          = 'N':  Non-unit triangular


          = 'U':  Unit triangular

N

          N is INTEGER


          The order of the matrix A.  N >= 0.  When N = 0, SLANTP is


          set to zero.

AP

          AP is REAL array, dimension (N*(N+1)/2)


          The upper or lower triangular matrix A, packed columnwise in


          a linear array.  The j-th column of A is stored in the array


          AP as follows:


          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;


          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.


          Note that when DIAG = 'U', the elements of the array AP


          corresponding to the diagonal elements of the matrix A are


          not referenced, but are assumed to be one.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= N when NORM = 'I'; otherwise, WORK is not


          referenced.

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 SLANTR  returns the value of the one norm,  or the Frobenius norm, or


 the  infinity norm,  or the  element of  largest absolute value  of a


 trapezoidal or triangular matrix A.

SLANTR

    SLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'


             (


             ( norm1(A),         NORM = '1', 'O' or 'o'


             (


             ( normI(A),         NORM = 'I' or 'i'


             (


             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'


 where  norm1  denotes the  one norm of a matrix (maximum column sum),


 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and


 normF  denotes the  Frobenius norm of a matrix (square root of sum of


 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.

NORM

          NORM is CHARACTER*1


          Specifies the value to be returned in SLANTR as described


          above.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the matrix A is upper or lower trapezoidal.


          = 'U':  Upper trapezoidal


          = 'L':  Lower trapezoidal


          Note that A is triangular instead of trapezoidal if M = N.

DIAG

          DIAG is CHARACTER*1


          Specifies whether or not the matrix A has unit diagonal.


          = 'N':  Non-unit diagonal


          = 'U':  Unit diagonal

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0, and if


          UPLO = 'U', M <= N.  When M = 0, SLANTR is set to zero.

N

          N is INTEGER


          The number of columns of the matrix A.  N >= 0, and if


          UPLO = 'L', N <= M.  When N = 0, SLANTR is set to zero.

A

          A is REAL array, dimension (LDA,N)


          The trapezoidal matrix A (A is triangular if M = N).


          If UPLO = 'U', the leading m by n upper trapezoidal part of


          the array A contains the upper trapezoidal matrix, and the


          strictly lower triangular part of A is not referenced.


          If UPLO = 'L', the leading m by n lower trapezoidal part of


          the array A contains the lower trapezoidal matrix, and the


          strictly upper triangular part of A is not referenced.  Note


          that when DIAG = 'U', the diagonal elements of A are not


          referenced and are assumed to be one.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(M,1).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),


          where LWORK >= M when NORM = 'I'; otherwise, WORK is not


          referenced.

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 SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric


 matrix in standard form:


      [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]


      [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]


 where either


 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or


 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex


 conjugate eigenvalues.

A

          A is REAL

B

          B is REAL

C

          C is REAL

D

          D is REAL


          On entry, the elements of the input matrix.


          On exit, they are overwritten by the elements of the


          standardised Schur form.

RT1R

          RT1R is REAL

RT1I

          RT1I is REAL

RT2R

          RT2R is REAL

RT2I

          RT2I is REAL


          The real and imaginary parts of the eigenvalues. If the


          eigenvalues are a complex conjugate pair, RT1I > 0.

CS

          CS is REAL

SN

          SN is REAL


          Parameters of the rotation matrix.

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  Modified by V. Sima, Research Institute for Informatics, Bucharest,


  Romania, to reduce the risk of cancellation errors,


  when computing real eigenvalues, and to ensure, if possible, that


  abs(RT1R) >= abs(RT2R).

 Given two column vectors X and Y, let


                      A = ( X Y ).


 The subroutine first computes the QR factorization of A = Q*R,


 and then computes the SVD of the 2-by-2 upper triangular matrix R.


 The smaller singular value of R is returned in SSMIN, which is used


 as the measurement of the linear dependency of the vectors X and Y.

N

          N is INTEGER


          The length of the vectors X and Y.

X

          X is REAL array,


                         dimension (1+(N-1)*INCX)


          On entry, X contains the N-vector X.


          On exit, X is overwritten.

INCX

          INCX is INTEGER


          The increment between successive elements of X. INCX > 0.

Y

          Y is REAL array,


                         dimension (1+(N-1)*INCY)


          On entry, Y contains the N-vector Y.


          On exit, Y is overwritten.

INCY

          INCY is INTEGER


          The increment between successive elements of Y. INCY > 0.

SSMIN

          SSMIN is REAL


          The smallest singular value of the N-by-2 matrix A = ( X Y ).

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 SLAPMR rearranges the rows of the M by N matrix X as specified


 by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.


 If FORWRD = .TRUE.,  forward permutation:


      X(K(I),*) is moved X(I,*) for I = 1,2,...,M.


 If FORWRD = .FALSE., backward permutation:


      X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.

FORWRD

          FORWRD is LOGICAL


          = .TRUE., forward permutation


          = .FALSE., backward permutation

M

          M is INTEGER


          The number of rows of the matrix X. M >= 0.

N

          N is INTEGER


          The number of columns of the matrix X. N >= 0.

X

          X is REAL array, dimension (LDX,N)


          On entry, the M by N matrix X.


          On exit, X contains the permuted matrix X.

LDX

          LDX is INTEGER


          The leading dimension of the array X, LDX >= MAX(1,M).

K

          K is INTEGER array, dimension (M)


          On entry, K contains the permutation vector. K is used as


          internal workspace, but reset to its original value on


          output.

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 SLAPMT rearranges the columns of the M by N matrix X as specified


 by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.


 If FORWRD = .TRUE.,  forward permutation:


      X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.


 If FORWRD = .FALSE., backward permutation:


      X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

FORWRD

          FORWRD is LOGICAL


          = .TRUE., forward permutation


          = .FALSE., backward permutation

M

          M is INTEGER


          The number of rows of the matrix X. M >= 0.

N

          N is INTEGER


          The number of columns of the matrix X. N >= 0.

X

          X is REAL array, dimension (LDX,N)


          On entry, the M by N matrix X.


          On exit, X contains the permuted matrix X.

LDX

          LDX is INTEGER


          The leading dimension of the array X, LDX >= MAX(1,M).

K

          K is INTEGER array, dimension (N)


          On entry, K contains the permutation vector. K is used as


          internal workspace, but reset to its original value on


          output.

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 SLAQP2 computes a QR factorization with column pivoting of


 the block A(OFFSET+1:M,1:N).


 The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

M

          M is INTEGER


          The number of rows of the matrix A. M >= 0.

N

          N is INTEGER


          The number of columns of the matrix A. N >= 0.

OFFSET

          OFFSET is INTEGER


          The number of rows of the matrix A that must be pivoted


          but no factorized. OFFSET >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the M-by-N matrix A.


          On exit, the upper triangle of block A(OFFSET+1:M,1:N) is


          the triangular factor obtained; the elements in block


          A(OFFSET+1:M,1:N) below the diagonal, together with the


          array TAU, represent the orthogonal matrix Q as a product of


          elementary reflectors. Block A(1:OFFSET,1:N) has been


          accordingly pivoted, but no factorized.

LDA

          LDA is INTEGER


          The leading dimension of the array A. LDA >= max(1,M).

JPVT

          JPVT is INTEGER array, dimension (N)


          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted


          to the front of A*P (a leading column); if JPVT(i) = 0,


          the i-th column of A is a free column.


          On exit, if JPVT(i) = k, then the i-th column of A*P


          was the k-th column of A.

TAU

          TAU is REAL array, dimension (min(M,N))


          The scalar factors of the elementary reflectors.

VN1

          VN1 is REAL array, dimension (N)


          The vector with the partial column norms.

VN2

          VN2 is REAL array, dimension (N)


          The vector with the exact column norms.

WORK

          WORK is REAL array, dimension (N)

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G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA
Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

LAPACK Working Note 176

 SLAQPS computes a step of QR factorization with column pivoting


 of a real M-by-N matrix A by using Blas-3.  It tries to factorize


 NB columns from A starting from the row OFFSET+1, and updates all


 of the matrix with Blas-3 xGEMM.


 In some cases, due to catastrophic cancellations, it cannot


 factorize NB columns.  Hence, the actual number of factorized


 columns is returned in KB.


 Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

M

          M is INTEGER


          The number of rows of the matrix A. M >= 0.

N

          N is INTEGER


          The number of columns of the matrix A. N >= 0

OFFSET

          OFFSET is INTEGER


          The number of rows of A that have been factorized in


          previous steps.

NB

          NB is INTEGER


          The number of columns to factorize.

KB

          KB is INTEGER


          The number of columns actually factorized.

A

          A is REAL array, dimension (LDA,N)


          On entry, the M-by-N matrix A.


          On exit, block A(OFFSET+1:M,1:KB) is the triangular


          factor obtained and block A(1:OFFSET,1:N) has been


          accordingly pivoted, but no factorized.


          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has


          been updated.

LDA

          LDA is INTEGER


          The leading dimension of the array A. LDA >= max(1,M).

JPVT

          JPVT is INTEGER array, dimension (N)


          JPVT(I) = K <==> Column K of the full matrix A has been


          permuted into position I in AP.

TAU

          TAU is REAL array, dimension (KB)


          The scalar factors of the elementary reflectors.

VN1

          VN1 is REAL array, dimension (N)


          The vector with the partial column norms.

VN2

          VN2 is REAL array, dimension (N)


          The vector with the exact column norms.

AUXV

          AUXV is REAL array, dimension (NB)


          Auxiliary vector.

F

          F is REAL array, dimension (LDF,NB)


          Matrix F**T = L*Y**T*A.

LDF

          LDF is INTEGER


          The leading dimension of the array F. LDF >= max(1,N).

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G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

LAPACK Working Note 176

    SLAQR0 computes the eigenvalues of a Hessenberg matrix H


    and, optionally, the matrices T and Z from the Schur decomposition


    H = Z T Z**T, where T is an upper quasi-triangular matrix (the


    Schur form), and Z is the orthogonal matrix of Schur vectors.


    Optionally Z may be postmultiplied into an input orthogonal


    matrix Q so that this routine can give the Schur factorization


    of a matrix A which has been reduced to the Hessenberg form H


    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

WANTT

          WANTT is LOGICAL


          = .TRUE. : the full Schur form T is required;


          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL


          = .TRUE. : the matrix of Schur vectors Z is required;


          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER


           The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


           It is assumed that H is already upper triangular in rows


           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,


           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a


           previous call to SGEBAL, and then passed to SGEHRD when the


           matrix output by SGEBAL is reduced to Hessenberg form.


           Otherwise, ILO and IHI should be set to 1 and N,


           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.


           If N = 0, then ILO = 1 and IHI = 0.

H

          H is REAL array, dimension (LDH,N)


           On entry, the upper Hessenberg matrix H.


           On exit, if INFO = 0 and WANTT is .TRUE., then H contains


           the upper quasi-triangular matrix T from the Schur


           decomposition (the Schur form); 2-by-2 diagonal blocks


           (corresponding to complex conjugate pairs of eigenvalues)


           are returned in standard form, with H(i,i) = H(i+1,i+1)


           and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is


           .FALSE., then the contents of H are unspecified on exit.


           (The output value of H when INFO > 0 is given under the


           description of INFO below.)


           This subroutine may explicitly set H(i,j) = 0 for i > j and


           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER


           The leading dimension of the array H. LDH >= max(1,N).

WR

          WR is REAL array, dimension (IHI)

WI

          WI is REAL array, dimension (IHI)


           The real and imaginary parts, respectively, of the computed


           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)


           and WI(ILO:IHI). If two eigenvalues are computed as a


           complex conjugate pair, they are stored in consecutive


           elements of WR and WI, say the i-th and (i+1)th, with


           WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then


           the eigenvalues are stored in the same order as on the


           diagonal of the Schur form returned in H, with


           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal


           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and


           WI(i+1) = -WI(i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


           Specify the rows of Z to which transformations must be


           applied if WANTZ is .TRUE..


           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is REAL array, dimension (LDZ,IHI)


           If WANTZ is .FALSE., then Z is not referenced.


           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is


           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the


           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).


           (The output value of Z when INFO > 0 is given under


           the description of INFO below.)

LDZ

          LDZ is INTEGER


           The leading dimension of the array Z.  if WANTZ is .TRUE.


           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

WORK

          WORK is REAL array, dimension LWORK


           On exit, if LWORK = -1, WORK(1) returns an estimate of


           the optimal value for LWORK.

LWORK

          LWORK is INTEGER


           The dimension of the array WORK.  LWORK >= max(1,N)


           is sufficient, but LWORK typically as large as 6*N may


           be required for optimal performance.  A workspace query


           to determine the optimal workspace size is recommended.


           If LWORK = -1, then SLAQR0 does a workspace query.


           In this case, SLAQR0 checks the input parameters and


           estimates the optimal workspace size for the given


           values of N, ILO and IHI.  The estimate is returned


           in WORK(1).  No error message related to LWORK is


           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER


             = 0:  successful exit


             > 0:  if INFO = i, SLAQR0 failed to compute all of


                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR


                and WI contain those eigenvalues which have been


                successfully computed.  (Failures are rare.)


                If INFO > 0 and WANT is .FALSE., then on exit,


                the remaining unconverged eigenvalues are the eigen-


                values of the upper Hessenberg matrix rows and


                columns ILO through INFO of the final, output


                value of H.


                If INFO > 0 and WANTT is .TRUE., then on exit


           (*)  (initial value of H)*U  = U*(final value of H)


                where U is an orthogonal matrix.  The final


                value of H is upper Hessenberg and quasi-triangular


                in rows and columns INFO+1 through IHI.


                If INFO > 0 and WANTZ is .TRUE., then on exit


                  (final value of Z(ILO:IHI,ILOZ:IHIZ)


                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U


                where U is the orthogonal matrix in (*) (regard-


                less of the value of WANTT.)


                If INFO > 0 and WANTZ is .FALSE., then Z is not


                accessed.

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Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR


  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3


  Performance, SIAM Journal of Matrix Analysis, volume 23, pages


  929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

      Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a


      scalar multiple of the first column of the product


      (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)


      scaling to avoid overflows and most underflows. It


      is assumed that either


              1) sr1 = sr2 and si1 = -si2


          or


              2) si1 = si2 = 0.


      This is useful for starting double implicit shift bulges


      in the QR algorithm.

N

          N is INTEGER


              Order of the matrix H. N must be either 2 or 3.

H

          H is REAL array, dimension (LDH,N)


              The 2-by-2 or 3-by-3 matrix H in (*).

LDH

          LDH is INTEGER


              The leading dimension of H as declared in


              the calling procedure.  LDH >= N

SR1

          SR1 is REAL

SI1

          SI1 is REAL

SR2

          SR2 is REAL

SI2

          SI2 is REAL


              The shifts in (*).

V

          V is REAL array, dimension (N)


              A scalar multiple of the first column of the


              matrix K in (*).

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    SLAQR2 is identical to SLAQR3 except that it avoids


    recursion by calling SLAHQR instead of SLAQR4.


    Aggressive early deflation:


    This subroutine accepts as input an upper Hessenberg matrix


    H and performs an orthogonal similarity transformation


    designed to detect and deflate fully converged eigenvalues from


    a trailing principal submatrix.  On output H has been over-


    written by a new Hessenberg matrix that is a perturbation of


    an orthogonal similarity transformation of H.  It is to be


    hoped that the final version of H has many zero subdiagonal


    entries.

WANTT

          WANTT is LOGICAL


          If .TRUE., then the Hessenberg matrix H is fully updated


          so that the quasi-triangular Schur factor may be


          computed (in cooperation with the calling subroutine).


          If .FALSE., then only enough of H is updated to preserve


          the eigenvalues.

WANTZ

          WANTZ is LOGICAL


          If .TRUE., then the orthogonal matrix Z is updated so


          so that the orthogonal Schur factor may be computed


          (in cooperation with the calling subroutine).


          If .FALSE., then Z is not referenced.

N

          N is INTEGER


          The order of the matrix H and (if WANTZ is .TRUE.) the


          order of the orthogonal matrix Z.

KTOP

          KTOP is INTEGER


          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.


          KBOT and KTOP together determine an isolated block


          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER


          It is assumed without a check that either


          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together


          determine an isolated block along the diagonal of the


          Hessenberg matrix.

NW

          NW is INTEGER


          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

H

          H is REAL array, dimension (LDH,N)


          On input the initial N-by-N section of H stores the


          Hessenberg matrix undergoing aggressive early deflation.


          On output H has been transformed by an orthogonal


          similarity transformation, perturbed, and the returned


          to Hessenberg form that (it is to be hoped) has some


          zero subdiagonal entries.

LDH

          LDH is INTEGER


          Leading dimension of H just as declared in the calling


          subroutine.  N <= LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


          Specify the rows of Z to which transformations must be


          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

Z

          Z is REAL array, dimension (LDZ,N)


          IF WANTZ is .TRUE., then on output, the orthogonal


          similarity transformation mentioned above has been


          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.


          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER


          The leading dimension of Z just as declared in the


          calling subroutine.  1 <= LDZ.

NS

          NS is INTEGER


          The number of unconverged (ie approximate) eigenvalues


          returned in SR and SI that may be used as shifts by the


          calling subroutine.

ND

          ND is INTEGER


          The number of converged eigenvalues uncovered by this


          subroutine.

SR

          SR is REAL array, dimension (KBOT)

SI

          SI is REAL array, dimension (KBOT)


          On output, the real and imaginary parts of approximate


          eigenvalues that may be used for shifts are stored in


          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and


          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.


          The real and imaginary parts of converged eigenvalues


          are stored in SR(KBOT-ND+1) through SR(KBOT) and


          SI(KBOT-ND+1) through SI(KBOT), respectively.

V

          V is REAL array, dimension (LDV,NW)


          An NW-by-NW work array.

LDV

          LDV is INTEGER


          The leading dimension of V just as declared in the


          calling subroutine.  NW <= LDV

NH

          NH is INTEGER


          The number of columns of T.  NH >= NW.

T

          T is REAL array, dimension (LDT,NW)

LDT

          LDT is INTEGER


          The leading dimension of T just as declared in the


          calling subroutine.  NW <= LDT

NV

          NV is INTEGER


          The number of rows of work array WV available for


          workspace.  NV >= NW.

WV

          WV is REAL array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER


          The leading dimension of W just as declared in the


          calling subroutine.  NW <= LDV

WORK

          WORK is REAL array, dimension (LWORK)


          On exit, WORK(1) is set to an estimate of the optimal value


          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER


          The dimension of the work array WORK.  LWORK = 2*NW


          suffices, but greater efficiency may result from larger


          values of LWORK.


          If LWORK = -1, then a workspace query is assumed; SLAQR2


          only estimates the optimal workspace size for the given


          values of N, NW, KTOP and KBOT.  The estimate is returned


          in WORK(1).  No error message related to LWORK is issued


          by XERBLA.  Neither H nor Z are accessed.

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Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

    Aggressive early deflation:


    SLAQR3 accepts as input an upper Hessenberg matrix


    H and performs an orthogonal similarity transformation


    designed to detect and deflate fully converged eigenvalues from


    a trailing principal submatrix.  On output H has been over-


    written by a new Hessenberg matrix that is a perturbation of


    an orthogonal similarity transformation of H.  It is to be


    hoped that the final version of H has many zero subdiagonal


    entries.

WANTT

          WANTT is LOGICAL


          If .TRUE., then the Hessenberg matrix H is fully updated


          so that the quasi-triangular Schur factor may be


          computed (in cooperation with the calling subroutine).


          If .FALSE., then only enough of H is updated to preserve


          the eigenvalues.

WANTZ

          WANTZ is LOGICAL


          If .TRUE., then the orthogonal matrix Z is updated so


          so that the orthogonal Schur factor may be computed


          (in cooperation with the calling subroutine).


          If .FALSE., then Z is not referenced.

N

          N is INTEGER


          The order of the matrix H and (if WANTZ is .TRUE.) the


          order of the orthogonal matrix Z.

KTOP

          KTOP is INTEGER


          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.


          KBOT and KTOP together determine an isolated block


          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER


          It is assumed without a check that either


          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together


          determine an isolated block along the diagonal of the


          Hessenberg matrix.

NW

          NW is INTEGER


          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

H

          H is REAL array, dimension (LDH,N)


          On input the initial N-by-N section of H stores the


          Hessenberg matrix undergoing aggressive early deflation.


          On output H has been transformed by an orthogonal


          similarity transformation, perturbed, and the returned


          to Hessenberg form that (it is to be hoped) has some


          zero subdiagonal entries.

LDH

          LDH is INTEGER


          Leading dimension of H just as declared in the calling


          subroutine.  N <= LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


          Specify the rows of Z to which transformations must be


          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

Z

          Z is REAL array, dimension (LDZ,N)


          IF WANTZ is .TRUE., then on output, the orthogonal


          similarity transformation mentioned above has been


          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.


          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER


          The leading dimension of Z just as declared in the


          calling subroutine.  1 <= LDZ.

NS

          NS is INTEGER


          The number of unconverged (ie approximate) eigenvalues


          returned in SR and SI that may be used as shifts by the


          calling subroutine.

ND

          ND is INTEGER


          The number of converged eigenvalues uncovered by this


          subroutine.

SR

          SR is REAL array, dimension (KBOT)

SI

          SI is REAL array, dimension (KBOT)


          On output, the real and imaginary parts of approximate


          eigenvalues that may be used for shifts are stored in


          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and


          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.


          The real and imaginary parts of converged eigenvalues


          are stored in SR(KBOT-ND+1) through SR(KBOT) and


          SI(KBOT-ND+1) through SI(KBOT), respectively.

V

          V is REAL array, dimension (LDV,NW)


          An NW-by-NW work array.

LDV

          LDV is INTEGER


          The leading dimension of V just as declared in the


          calling subroutine.  NW <= LDV

NH

          NH is INTEGER


          The number of columns of T.  NH >= NW.

T

          T is REAL array, dimension (LDT,NW)

LDT

          LDT is INTEGER


          The leading dimension of T just as declared in the


          calling subroutine.  NW <= LDT

NV

          NV is INTEGER


          The number of rows of work array WV available for


          workspace.  NV >= NW.

WV

          WV is REAL array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER


          The leading dimension of W just as declared in the


          calling subroutine.  NW <= LDV

WORK

          WORK is REAL array, dimension (LWORK)


          On exit, WORK(1) is set to an estimate of the optimal value


          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER


          The dimension of the work array WORK.  LWORK = 2*NW


          suffices, but greater efficiency may result from larger


          values of LWORK.


          If LWORK = -1, then a workspace query is assumed; SLAQR3


          only estimates the optimal workspace size for the given


          values of N, NW, KTOP and KBOT.  The estimate is returned


          in WORK(1).  No error message related to LWORK is issued


          by XERBLA.  Neither H nor Z are accessed.

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    SLAQR4 implements one level of recursion for SLAQR0.


    It is a complete implementation of the small bulge multi-shift


    QR algorithm.  It may be called by SLAQR0 and, for large enough


    deflation window size, it may be called by SLAQR3.  This


    subroutine is identical to SLAQR0 except that it calls SLAQR2


    instead of SLAQR3.


    SLAQR4 computes the eigenvalues of a Hessenberg matrix H


    and, optionally, the matrices T and Z from the Schur decomposition


    H = Z T Z**T, where T is an upper quasi-triangular matrix (the


    Schur form), and Z is the orthogonal matrix of Schur vectors.


    Optionally Z may be postmultiplied into an input orthogonal


    matrix Q so that this routine can give the Schur factorization


    of a matrix A which has been reduced to the Hessenberg form H


    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

WANTT

          WANTT is LOGICAL


          = .TRUE. : the full Schur form T is required;


          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL


          = .TRUE. : the matrix of Schur vectors Z is required;


          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER


           The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


           It is assumed that H is already upper triangular in rows


           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,


           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a


           previous call to SGEBAL, and then passed to SGEHRD when the


           matrix output by SGEBAL is reduced to Hessenberg form.


           Otherwise, ILO and IHI should be set to 1 and N,


           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.


           If N = 0, then ILO = 1 and IHI = 0.

H

          H is REAL array, dimension (LDH,N)


           On entry, the upper Hessenberg matrix H.


           On exit, if INFO = 0 and WANTT is .TRUE., then H contains


           the upper quasi-triangular matrix T from the Schur


           decomposition (the Schur form); 2-by-2 diagonal blocks


           (corresponding to complex conjugate pairs of eigenvalues)


           are returned in standard form, with H(i,i) = H(i+1,i+1)


           and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is


           .FALSE., then the contents of H are unspecified on exit.


           (The output value of H when INFO > 0 is given under the


           description of INFO below.)


           This subroutine may explicitly set H(i,j) = 0 for i > j and


           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER


           The leading dimension of the array H. LDH >= max(1,N).

WR

          WR is REAL array, dimension (IHI)

WI

          WI is REAL array, dimension (IHI)


           The real and imaginary parts, respectively, of the computed


           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)


           and WI(ILO:IHI). If two eigenvalues are computed as a


           complex conjugate pair, they are stored in consecutive


           elements of WR and WI, say the i-th and (i+1)th, with


           WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then


           the eigenvalues are stored in the same order as on the


           diagonal of the Schur form returned in H, with


           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal


           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and


           WI(i+1) = -WI(i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


           Specify the rows of Z to which transformations must be


           applied if WANTZ is .TRUE..


           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is REAL array, dimension (LDZ,IHI)


           If WANTZ is .FALSE., then Z is not referenced.


           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is


           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the


           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).


           (The output value of Z when INFO > 0 is given under


           the description of INFO below.)

LDZ

          LDZ is INTEGER


           The leading dimension of the array Z.  if WANTZ is .TRUE.


           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

WORK

          WORK is REAL array, dimension LWORK


           On exit, if LWORK = -1, WORK(1) returns an estimate of


           the optimal value for LWORK.

LWORK

          LWORK is INTEGER


           The dimension of the array WORK.  LWORK >= max(1,N)


           is sufficient, but LWORK typically as large as 6*N may


           be required for optimal performance.  A workspace query


           to determine the optimal workspace size is recommended.


           If LWORK = -1, then SLAQR4 does a workspace query.


           In this case, SLAQR4 checks the input parameters and


           estimates the optimal workspace size for the given


           values of N, ILO and IHI.  The estimate is returned


           in WORK(1).  No error message related to LWORK is


           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER


 \verbatim


          INFO is INTEGER


             = 0:  successful exit


             > 0:  if INFO = i, SLAQR4 failed to compute all of


                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR


                and WI contain those eigenvalues which have been


                successfully computed.  (Failures are rare.)


                If INFO > 0 and WANT is .FALSE., then on exit,


                the remaining unconverged eigenvalues are the eigen-


                values of the upper Hessenberg matrix rows and


                columns ILO through INFO of the final, output


                value of H.


                If INFO > 0 and WANTT is .TRUE., then on exit


           (*)  (initial value of H)*U  = U*(final value of H)


                where U is a orthogonal matrix.  The final


                value of  H is upper Hessenberg and triangular in


                rows and columns INFO+1 through IHI.


                If INFO > 0 and WANTZ is .TRUE., then on exit


                  (final value of Z(ILO:IHI,ILOZ:IHIZ)


                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U


                where U is the orthogonal matrix in (*) (regard-


                less of the value of WANTT.)


                If INFO > 0 and WANTZ is .FALSE., then Z is not


                accessed.

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NAG Ltd.

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR


  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3


  Performance, SIAM Journal of Matrix Analysis, volume 23, pages


  929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

    SLAQR5, called by SLAQR0, performs a


    single small-bulge multi-shift QR sweep.

WANTT

          WANTT is LOGICAL


             WANTT = .true. if the quasi-triangular Schur factor


             is being computed.  WANTT is set to .false. otherwise.

WANTZ

          WANTZ is LOGICAL


             WANTZ = .true. if the orthogonal Schur factor is being


             computed.  WANTZ is set to .false. otherwise.

KACC22

          KACC22 is INTEGER with value 0, 1, or 2.


             Specifies the computation mode of far-from-diagonal


             orthogonal updates.


        = 0: SLAQR5 does not accumulate reflections and does not


             use matrix-matrix multiply to update far-from-diagonal


             matrix entries.


        = 1: SLAQR5 accumulates reflections and uses matrix-matrix


             multiply to update the far-from-diagonal matrix entries.


        = 2: Same as KACC22 = 1. This option used to enable exploiting


             the 2-by-2 structure during matrix multiplications, but


             this is no longer supported.

N

          N is INTEGER


             N is the order of the Hessenberg matrix H upon which this


             subroutine operates.

KTOP

          KTOP is INTEGER

KBOT

          KBOT is INTEGER


             These are the first and last rows and columns of an


             isolated diagonal block upon which the QR sweep is to be


             applied. It is assumed without a check that


                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0


             and


                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS

          NSHFTS is INTEGER


             NSHFTS gives the number of simultaneous shifts.  NSHFTS


             must be positive and even.

SR

          SR is REAL array, dimension (NSHFTS)

SI

          SI is REAL array, dimension (NSHFTS)


             SR contains the real parts and SI contains the imaginary


             parts of the NSHFTS shifts of origin that define the


             multi-shift QR sweep.  On output SR and SI may be


             reordered.

H

          H is REAL array, dimension (LDH,N)


             On input H contains a Hessenberg matrix.  On output a


             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied


             to the isolated diagonal block in rows and columns KTOP


             through KBOT.

LDH

          LDH is INTEGER


             LDH is the leading dimension of H just as declared in the


             calling procedure.  LDH >= MAX(1,N).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


             Specify the rows of Z to which transformations must be


             applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N

Z

          Z is REAL array, dimension (LDZ,IHIZ)


             If WANTZ = .TRUE., then the QR Sweep orthogonal


             similarity transformation is accumulated into


             Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.


             If WANTZ = .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER


             LDA is the leading dimension of Z just as declared in


             the calling procedure. LDZ >= N.

V

          V is REAL array, dimension (LDV,NSHFTS/2)

LDV

          LDV is INTEGER


             LDV is the leading dimension of V as declared in the


             calling procedure.  LDV >= 3.

U

          U is REAL array, dimension (LDU,2*NSHFTS)

LDU

          LDU is INTEGER


             LDU is the leading dimension of U just as declared in the


             in the calling subroutine.  LDU >= 2*NSHFTS.

NV

          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV >= 1.

WV

          WV is REAL array, dimension (LDWV,2*NSHFTS)

LDWV

          LDWV is INTEGER


             LDWV is the leading dimension of WV as declared in the


             in the calling subroutine.  LDWV >= NV.

NH

          NH is INTEGER


             NH is the number of columns in array WH available for


             workspace. NH >= 1.

WH

          WH is REAL array, dimension (LDWH,NH)

LDWH

          LDWH is INTEGER


             Leading dimension of WH just as declared in the


             calling procedure.  LDWH >= 2*NSHFTS.

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NAG Ltd.

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

 SLAQSB equilibrates a symmetric band matrix A using the scaling


 factors in the vector S.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          symmetric matrix A is stored.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER


          The number of super-diagonals of the matrix A if UPLO = 'U',


          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is REAL array, dimension (LDAB,N)


          On entry, the upper or lower triangle of the symmetric band


          matrix A, stored in the first KD+1 rows of the array.  The


          j-th column of A is stored in the j-th column of the array AB


          as follows:


          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;


          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).


          On exit, if INFO = 0, the triangular factor U or L from the


          Cholesky factorization A = U**T*U or A = L*L**T of the band


          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= KD+1.

S

          S is REAL array, dimension (N)


          The scale factors for A.

SCOND

          SCOND is REAL


          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is REAL


          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1


          Specifies whether or not equilibration was done.


          = 'N':  No equilibration.


          = 'Y':  Equilibration was done, i.e., A has been replaced by


                  diag(S) * A * diag(S).

  THRESH is a threshold value used to decide if scaling should be done


  based on the ratio of the scaling factors.  If SCOND < THRESH,


  scaling is done.


  LARGE and SMALL are threshold values used to decide if scaling should


  be done based on the absolute size of the largest matrix element.


  If AMAX > LARGE or AMAX < SMALL, scaling is done.

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 SLAQSP equilibrates a symmetric matrix A using the scaling factors


 in the vector S.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          symmetric matrix A is stored.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

AP

          AP is REAL array, dimension (N*(N+1)/2)


          On entry, the upper or lower triangle of the symmetric matrix


          A, packed columnwise in a linear array.  The j-th column of A


          is stored in the array AP as follows:


          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;


          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.


          On exit, the equilibrated matrix:  diag(S) * A * diag(S), in


          the same storage format as A.

S

          S is REAL array, dimension (N)


          The scale factors for A.

SCOND

          SCOND is REAL


          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is REAL


          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1


          Specifies whether or not equilibration was done.


          = 'N':  No equilibration.


          = 'Y':  Equilibration was done, i.e., A has been replaced by


                  diag(S) * A * diag(S).

  THRESH is a threshold value used to decide if scaling should be done


  based on the ratio of the scaling factors.  If SCOND < THRESH,


  scaling is done.


  LARGE and SMALL are threshold values used to decide if scaling should


  be done based on the absolute size of the largest matrix element.


  If AMAX > LARGE or AMAX < SMALL, scaling is done.

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 SLAQTR solves the real quasi-triangular system


              op(T)*p = scale*c,               if LREAL = .TRUE.


 or the complex quasi-triangular systems


            op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.


 in real arithmetic, where T is upper quasi-triangular.


 If LREAL = .FALSE., then the first diagonal block of T must be


 1 by 1, B is the specially structured matrix


                B = [ b(1) b(2) ... b(n) ]


                    [       w            ]


                    [           w        ]


                    [              .     ]


                    [                 w  ]


 op(A) = A or A**T, A**T denotes the transpose of


 matrix A.


 On input, X = [ c ].  On output, X = [ p ].


               [ d ]                  [ q ]


 This subroutine is designed for the condition number estimation


 in routine STRSNA.

LTRAN

          LTRAN is LOGICAL


          On entry, LTRAN specifies the option of conjugate transpose:


             = .FALSE.,    op(T+i*B) = T+i*B,


             = .TRUE.,     op(T+i*B) = (T+i*B)**T.

LREAL

          LREAL is LOGICAL


          On entry, LREAL specifies the input matrix structure:


             = .FALSE.,    the input is complex


             = .TRUE.,     the input is real

N

          N is INTEGER


          On entry, N specifies the order of T+i*B. N >= 0.

T

          T is REAL array, dimension (LDT,N)


          On entry, T contains a matrix in Schur canonical form.


          If LREAL = .FALSE., then the first diagonal block of T must


          be 1 by 1.

LDT

          LDT is INTEGER


          The leading dimension of the matrix T. LDT >= max(1,N).

B

          B is REAL array, dimension (N)


          On entry, B contains the elements to form the matrix


          B as described above.


          If LREAL = .TRUE., B is not referenced.

W

          W is REAL


          On entry, W is the diagonal element of the matrix B.


          If LREAL = .TRUE., W is not referenced.

SCALE

          SCALE is REAL


          On exit, SCALE is the scale factor.

X

          X is REAL array, dimension (2*N)


          On entry, X contains the right hand side of the system.


          On exit, X is overwritten by the solution.

WORK

          WORK is REAL array, dimension (N)

INFO

          INFO is INTEGER


          On exit, INFO is set to


             0: successful exit.


               1: the some diagonal 1 by 1 block has been perturbed by


                  a small number SMIN to keep nonsingularity.


               2: the some diagonal 2 by 2 block has been perturbed by


                  a small number in SLALN2 to keep nonsingularity.


          NOTE: In the interests of speed, this routine does not


                check the inputs for errors.

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 SLAR1V computes the (scaled) r-th column of the inverse of


 the sumbmatrix in rows B1 through BN of the tridiagonal matrix


 L D L**T - sigma I. When sigma is close to an eigenvalue, the


 computed vector is an accurate eigenvector. Usually, r corresponds


 to the index where the eigenvector is largest in magnitude.


 The following steps accomplish this computation :


 (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,


 (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,


 (c) Computation of the diagonal elements of the inverse of


     L D L**T - sigma I by combining the above transforms, and choosing


     r as the index where the diagonal of the inverse is (one of the)


     largest in magnitude.


 (d) Computation of the (scaled) r-th column of the inverse using the


     twisted factorization obtained by combining the top part of the


     the stationary and the bottom part of the progressive transform.

N

          N is INTEGER


           The order of the matrix L D L**T.

B1

          B1 is INTEGER


           First index of the submatrix of L D L**T.

BN

          BN is INTEGER


           Last index of the submatrix of L D L**T.

LAMBDA

          LAMBDA is REAL


           The shift. In order to compute an accurate eigenvector,


           LAMBDA should be a good approximation to an eigenvalue


           of L D L**T.

L

          L is REAL array, dimension (N-1)


           The (n-1) subdiagonal elements of the unit bidiagonal matrix


           L, in elements 1 to N-1.

D

          D is REAL array, dimension (N)


           The n diagonal elements of the diagonal matrix D.

LD

          LD is REAL array, dimension (N-1)


           The n-1 elements L(i)*D(i).

LLD

          LLD is REAL array, dimension (N-1)


           The n-1 elements L(i)*L(i)*D(i).

PIVMIN

          PIVMIN is REAL


           The minimum pivot in the Sturm sequence.

GAPTOL

          GAPTOL is REAL


           Tolerance that indicates when eigenvector entries are negligible


           w.r.t. their contribution to the residual.

Z

          Z is REAL array, dimension (N)


           On input, all entries of Z must be set to 0.


           On output, Z contains the (scaled) r-th column of the


           inverse. The scaling is such that Z(R) equals 1.

WANTNC

          WANTNC is LOGICAL


           Specifies whether NEGCNT has to be computed.

NEGCNT

          NEGCNT is INTEGER


           If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin


           in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.

ZTZ

          ZTZ is REAL


           The square of the 2-norm of Z.

MINGMA

          MINGMA is REAL


           The reciprocal of the largest (in magnitude) diagonal


           element of the inverse of L D L**T - sigma I.

R

          R is INTEGER


           The twist index for the twisted factorization used to


           compute Z.


           On input, 0 <= R <= N. If R is input as 0, R is set to


           the index where (L D L**T - sigma I)^{-1} is largest


           in magnitude. If 1 <= R <= N, R is unchanged.


           On output, R contains the twist index used to compute Z.


           Ideally, R designates the position of the maximum entry in the


           eigenvector.

ISUPPZ

          ISUPPZ is INTEGER array, dimension (2)


           The support of the vector in Z, i.e., the vector Z is


           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

NRMINV

          NRMINV is REAL


           NRMINV = 1/SQRT( ZTZ )

RESID

          RESID is REAL


           The residual of the FP vector.


           RESID = ABS( MINGMA )/SQRT( ZTZ )

RQCORR

          RQCORR is REAL


           The Rayleigh Quotient correction to LAMBDA.


           RQCORR = MINGMA*TMP

WORK

          WORK is REAL array, dimension (4*N)

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Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

 SLAR2V applies a vector of real plane rotations from both sides to


 a sequence of 2-by-2 real symmetric matrices, defined by the elements


 of the vectors x, y and z. For i = 1,2,...,n


    ( x(i)  z(i) ) := (  c(i)  s(i) ) ( x(i)  z(i) ) ( c(i) -s(i) )


    ( z(i)  y(i) )    ( -s(i)  c(i) ) ( z(i)  y(i) ) ( s(i)  c(i) )

N

          N is INTEGER


          The number of plane rotations to be applied.

X

          X is REAL array,


                         dimension (1+(N-1)*INCX)


          The vector x.

Y

          Y is REAL array,


                         dimension (1+(N-1)*INCX)


          The vector y.

Z

          Z is REAL array,


                         dimension (1+(N-1)*INCX)


          The vector z.

INCX

          INCX is INTEGER


          The increment between elements of X, Y and Z. INCX > 0.

C

          C is REAL array, dimension (1+(N-1)*INCC)


          The cosines of the plane rotations.

S

          S is REAL array, dimension (1+(N-1)*INCC)


          The sines of the plane rotations.

INCC

          INCC is INTEGER


          The increment between elements of C and S. INCC > 0.

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 SLARF applies a real elementary reflector H to a real m by n matrix


 C, from either the left or the right. H is represented in the form


       H = I - tau * v * v**T


 where tau is a real scalar and v is a real vector.


 If tau = 0, then H is taken to be the unit matrix.

SIDE

          SIDE is CHARACTER*1


          = 'L': form  H * C


          = 'R': form  C * H

M

          M is INTEGER


          The number of rows of the matrix C.

N

          N is INTEGER


          The number of columns of the matrix C.

V

          V is REAL array, dimension


                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'


                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'


          The vector v in the representation of H. V is not used if


          TAU = 0.

INCV

          INCV is INTEGER


          The increment between elements of v. INCV <> 0.

TAU

          TAU is REAL


          The value tau in the representation of H.

C

          C is REAL array, dimension (LDC,N)


          On entry, the m by n matrix C.


          On exit, C is overwritten by the matrix H * C if SIDE = 'L',


          or C * H if SIDE = 'R'.

LDC

          LDC is INTEGER


          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is REAL array, dimension


                         (N) if SIDE = 'L'


                      or (M) if SIDE = 'R'

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 SLARFB applies a real block reflector H or its transpose H**T to a


 real m by n matrix C, from either the left or the right.

SIDE

          SIDE is CHARACTER*1


          = 'L': apply H or H**T from the Left


          = 'R': apply H or H**T from the Right

TRANS

          TRANS is CHARACTER*1


          = 'N': apply H (No transpose)


          = 'T': apply H**T (Transpose)

DIRECT

          DIRECT is CHARACTER*1


          Indicates how H is formed from a product of elementary


          reflectors


          = 'F': H = H(1) H(2) . . . H(k) (Forward)


          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1


          Indicates how the vectors which define the elementary


          reflectors are stored:


          = 'C': Columnwise


          = 'R': Rowwise

M

          M is INTEGER


          The number of rows of the matrix C.

N

          N is INTEGER


          The number of columns of the matrix C.

K

          K is INTEGER


          The order of the matrix T (= the number of elementary


          reflectors whose product defines the block reflector).


          If SIDE = 'L', M >= K >= 0;


          if SIDE = 'R', N >= K >= 0.

V

          V is REAL array, dimension


                                (LDV,K) if STOREV = 'C'


                                (LDV,M) if STOREV = 'R' and SIDE = 'L'


                                (LDV,N) if STOREV = 'R' and SIDE = 'R'


          The matrix V. See Further Details.

LDV

          LDV is INTEGER


          The leading dimension of the array V.


          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);


          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);


          if STOREV = 'R', LDV >= K.

T

          T is REAL array, dimension (LDT,K)


          The triangular k by k matrix T in the representation of the


          block reflector.

LDT

          LDT is INTEGER


          The leading dimension of the array T. LDT >= K.

C

          C is REAL array, dimension (LDC,N)


          On entry, the m by n matrix C.


          On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.

LDC

          LDC is INTEGER


          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is REAL array, dimension (LDWORK,K)

LDWORK

          LDWORK is INTEGER


          The leading dimension of the array WORK.


          If SIDE = 'L', LDWORK >= max(1,N);


          if SIDE = 'R', LDWORK >= max(1,M).

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  The shape of the matrix V and the storage of the vectors which define


  the H(i) is best illustrated by the following example with n = 5 and


  k = 3. The elements equal to 1 are not stored; the corresponding


  array elements are modified but restored on exit. The rest of the


  array is not used.


  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':


               V = (  1       )                 V = (  1 v1 v1 v1 v1 )


                   ( v1  1    )                     (     1 v2 v2 v2 )


                   ( v1 v2  1 )                     (        1 v3 v3 )


                   ( v1 v2 v3 )


                   ( v1 v2 v3 )


  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':


               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )


                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )


                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )


                   (     1 v3 )


                   (        1 )

 SLARFG generates a real elementary reflector H of order n, such


 that


       H * ( alpha ) = ( beta ),   H**T * H = I.


           (   x   )   (   0  )


 where alpha and beta are scalars, and x is an (n-1)-element real


 vector. H is represented in the form


       H = I - tau * ( 1 ) * ( 1 v**T ) ,


                     ( v )


 where tau is a real scalar and v is a real (n-1)-element


 vector.


 If the elements of x are all zero, then tau = 0 and H is taken to be


 the unit matrix.


 Otherwise  1 <= tau <= 2.

N

          N is INTEGER


          The order of the elementary reflector.

ALPHA

          ALPHA is REAL


          On entry, the value alpha.


          On exit, it is overwritten with the value beta.

X

          X is REAL array, dimension


                         (1+(N-2)*abs(INCX))


          On entry, the vector x.


          On exit, it is overwritten with the vector v.

INCX

          INCX is INTEGER


          The increment between elements of X. INCX > 0.

TAU

          TAU is REAL


          The value tau.

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 SLARFGP generates a real elementary reflector H of order n, such


 that


       H * ( alpha ) = ( beta ),   H**T * H = I.


           (   x   )   (   0  )


 where alpha and beta are scalars, beta is non-negative, and x is


 an (n-1)-element real vector.  H is represented in the form


       H = I - tau * ( 1 ) * ( 1 v**T ) ,


                     ( v )


 where tau is a real scalar and v is a real (n-1)-element


 vector.


 If the elements of x are all zero, then tau = 0 and H is taken to be


 the unit matrix.

N

          N is INTEGER


          The order of the elementary reflector.

ALPHA

          ALPHA is REAL


          On entry, the value alpha.


          On exit, it is overwritten with the value beta.

X

          X is REAL array, dimension


                         (1+(N-2)*abs(INCX))


          On entry, the vector x.


          On exit, it is overwritten with the vector v.

INCX

          INCX is INTEGER


          The increment between elements of X. INCX > 0.

TAU

          TAU is REAL


          The value tau.

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 SLARFT forms the triangular factor T of a real block reflector H


 of order n, which is defined as a product of k elementary reflectors.


 If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;


 If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.


 If STOREV = 'C', the vector which defines the elementary reflector


 H(i) is stored in the i-th column of the array V, and


    H  =  I - V * T * V**T


 If STOREV = 'R', the vector which defines the elementary reflector


 H(i) is stored in the i-th row of the array V, and


    H  =  I - V**T * T * V

DIRECT

          DIRECT is CHARACTER*1


          Specifies the order in which the elementary reflectors are


          multiplied to form the block reflector:


          = 'F': H = H(1) H(2) . . . H(k) (Forward)


          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1


          Specifies how the vectors which define the elementary


          reflectors are stored (see also Further Details):


          = 'C': columnwise


          = 'R': rowwise

N

          N is INTEGER


          The order of the block reflector H. N >= 0.

K

          K is INTEGER


          The order of the triangular factor T (= the number of


          elementary reflectors). K >= 1.

V

          V is REAL array, dimension


                               (LDV,K) if STOREV = 'C'


                               (LDV,N) if STOREV = 'R'


          The matrix V. See further details.

LDV

          LDV is INTEGER


          The leading dimension of the array V.


          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

TAU

          TAU is REAL array, dimension (K)


          TAU(i) must contain the scalar factor of the elementary


          reflector H(i).

T

          T is REAL array, dimension (LDT,K)


          The k by k triangular factor T of the block reflector.


          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is


          lower triangular. The rest of the array is not used.

LDT

          LDT is INTEGER


          The leading dimension of the array T. LDT >= K.

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  The shape of the matrix V and the storage of the vectors which define


  the H(i) is best illustrated by the following example with n = 5 and


  k = 3. The elements equal to 1 are not stored.


  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':


               V = (  1       )                 V = (  1 v1 v1 v1 v1 )


                   ( v1  1    )                     (     1 v2 v2 v2 )


                   ( v1 v2  1 )                     (        1 v3 v3 )


                   ( v1 v2 v3 )


                   ( v1 v2 v3 )


  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':


               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )


                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )


                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )


                   (     1 v3 )


                   (        1 )

 SLARFX applies a real elementary reflector H to a real m by n


 matrix C, from either the left or the right. H is represented in the


 form


       H = I - tau * v * v**T


 where tau is a real scalar and v is a real vector.


 If tau = 0, then H is taken to be the unit matrix


 This version uses inline code if H has order < 11.

SIDE

          SIDE is CHARACTER*1


          = 'L': form  H * C


          = 'R': form  C * H

M

          M is INTEGER


          The number of rows of the matrix C.

N

          N is INTEGER


          The number of columns of the matrix C.

V

          V is REAL array, dimension (M) if SIDE = 'L'


                                     or (N) if SIDE = 'R'


          The vector v in the representation of H.

TAU

          TAU is REAL


          The value tau in the representation of H.

C

          C is REAL array, dimension (LDC,N)


          On entry, the m by n matrix C.


          On exit, C is overwritten by the matrix H * C if SIDE = 'L',


          or C * H if SIDE = 'R'.

LDC

          LDC is INTEGER


          The leading dimension of the array C. LDC >= (1,M).

WORK

          WORK is REAL array, dimension


                      (N) if SIDE = 'L'


                      or (M) if SIDE = 'R'


          WORK is not referenced if H has order < 11.

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 SLARFY applies an elementary reflector, or Householder matrix, H,


 to an n x n symmetric matrix C, from both the left and the right.


 H is represented in the form


    H = I - tau * v * v'


 where  tau  is a scalar and  v  is a vector.


 If  tau  is  zero, then  H  is taken to be the unit matrix.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          symmetric matrix C is stored.


          = 'U':  Upper triangle


          = 'L':  Lower triangle

N

          N is INTEGER


          The number of rows and columns of the matrix C.  N >= 0.

V

          V is REAL array, dimension


                  (1 + (N-1)*abs(INCV))


          The vector v as described above.

INCV

          INCV is INTEGER


          The increment between successive elements of v.  INCV must


          not be zero.

TAU

          TAU is REAL


          The value tau as described above.

C

          C is REAL array, dimension (LDC, N)


          On entry, the matrix C.


          On exit, C is overwritten by H * C * H'.

LDC

          LDC is INTEGER


          The leading dimension of the array C.  LDC >= max( 1, N ).

WORK

          WORK is REAL array, dimension (N)

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 SLARGV generates a vector of real plane rotations, determined by


 elements of the real vectors x and y. For i = 1,2,...,n


    (  c(i)  s(i) ) ( x(i) ) = ( a(i) )


    ( -s(i)  c(i) ) ( y(i) ) = (   0  )

N

          N is INTEGER


          The number of plane rotations to be generated.

X

          X is REAL array,


                         dimension (1+(N-1)*INCX)


          On entry, the vector x.


          On exit, x(i) is overwritten by a(i), for i = 1,...,n.

INCX

          INCX is INTEGER


          The increment between elements of X. INCX > 0.

Y

          Y is REAL array,


                         dimension (1+(N-1)*INCY)


          On entry, the vector y.


          On exit, the sines of the plane rotations.

INCY

          INCY is INTEGER


          The increment between elements of Y. INCY > 0.

C

          C is REAL array, dimension (1+(N-1)*INCC)


          The cosines of the plane rotations.

INCC

          INCC is INTEGER


          The increment between elements of C. INCC > 0.

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 SLARRV computes the eigenvectors of the tridiagonal matrix


 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.


 The input eigenvalues should have been computed by SLARRE.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

VL

          VL is REAL


          Lower bound of the interval that contains the desired


          eigenvalues. VL < VU. Needed to compute gaps on the left or right


          end of the extremal eigenvalues in the desired RANGE.

VU

          VU is REAL


          Upper bound of the interval that contains the desired


          eigenvalues. VL < VU. 


          Note: VU is currently not used by this implementation of SLARRV, VU is


          passed to SLARRV because it could be used compute gaps on the right end


          of the extremal eigenvalues. However, with not much initial accuracy in


          LAMBDA and VU, the formula can lead to an overestimation of the right gap


          and thus to inadequately early RQI 'convergence'. This is currently


          prevented this by forcing a small right gap. And so it turns out that VU


          is currently not used by this implementation of SLARRV.

D

          D is REAL array, dimension (N)


          On entry, the N diagonal elements of the diagonal matrix D.


          On exit, D may be overwritten.

L

          L is REAL array, dimension (N)


          On entry, the (N-1) subdiagonal elements of the unit


          bidiagonal matrix L are in elements 1 to N-1 of L


          (if the matrix is not split.) At the end of each block


          is stored the corresponding shift as given by SLARRE.


          On exit, L is overwritten.

PIVMIN

          PIVMIN is REAL


          The minimum pivot allowed in the Sturm sequence.

ISPLIT

          ISPLIT is INTEGER array, dimension (N)


          The splitting points, at which T breaks up into blocks.


          The first block consists of rows/columns 1 to


          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1


          through ISPLIT( 2 ), etc.

M

          M is INTEGER


          The total number of input eigenvalues.  0 <= M <= N.

DOL

          DOL is INTEGER

DOU

          DOU is INTEGER


          If the user wants to compute only selected eigenvectors from all


          the eigenvalues supplied, he can specify an index range DOL:DOU.


          Or else the setting DOL=1, DOU=M should be applied.


          Note that DOL and DOU refer to the order in which the eigenvalues


          are stored in W.


          If the user wants to compute only selected eigenpairs, then


          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the


          computed eigenvectors. All other columns of Z are set to zero.

MINRGP

          MINRGP is REAL

RTOL1

          RTOL1 is REAL

RTOL2

          RTOL2 is REAL


           Parameters for bisection.


           An interval [LEFT,RIGHT] has converged if


           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

W

          W is REAL array, dimension (N)


          The first M elements of W contain the APPROXIMATE eigenvalues for


          which eigenvectors are to be computed.  The eigenvalues


          should be grouped by split-off block and ordered from


          smallest to largest within the block ( The output array


          W from SLARRE is expected here ). Furthermore, they are with


          respect to the shift of the corresponding root representation


          for their block. On exit, W holds the eigenvalues of the


          UNshifted matrix.

WERR

          WERR is REAL array, dimension (N)


          The first M elements contain the semiwidth of the uncertainty


          interval of the corresponding eigenvalue in W

WGAP

          WGAP is REAL array, dimension (N)


          The separation from the right neighbor eigenvalue in W.

IBLOCK

          IBLOCK is INTEGER array, dimension (N)


          The indices of the blocks (submatrices) associated with the


          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue


          W(i) belongs to the first block from the top, =2 if W(i)


          belongs to the second block, etc.

INDEXW

          INDEXW is INTEGER array, dimension (N)


          The indices of the eigenvalues within each block (submatrix);


          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the


          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

GERS

          GERS is REAL array, dimension (2*N)


          The N Gerschgorin intervals (the i-th Gerschgorin interval


          is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should


          be computed from the original UNshifted matrix.

Z

          Z is REAL array, dimension (LDZ, max(1,M) )


          If INFO = 0, the first M columns of Z contain the


          orthonormal eigenvectors of the matrix T


          corresponding to the input eigenvalues, with the i-th


          column of Z holding the eigenvector associated with W(i).


          Note: the user must ensure that at least max(1,M) columns are


          supplied in the array Z.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )


          The support of the eigenvectors in Z, i.e., the indices


          indicating the nonzero elements in Z. The I-th eigenvector


          is nonzero only in elements ISUPPZ( 2*I-1 ) through


          ISUPPZ( 2*I ).

WORK

          WORK is REAL array, dimension (12*N)

IWORK

          IWORK is INTEGER array, dimension (7*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          > 0:  A problem occurred in SLARRV.


          < 0:  One of the called subroutines signaled an internal problem.


                Needs inspection of the corresponding parameter IINFO


                for further information.


          =-1:  Problem in SLARRB when refining a child's eigenvalues.


          =-2:  Problem in SLARRF when computing the RRR of a child.


                When a child is inside a tight cluster, it can be difficult


                to find an RRR. A partial remedy from the user's point of


                view is to make the parameter MINRGP smaller and recompile.


                However, as the orthogonality of the computed vectors is


                proportional to 1/MINRGP, the user should be aware that


                he might be trading in precision when he decreases MINRGP.


          =-3:  Problem in SLARRB when refining a single eigenvalue


                after the Rayleigh correction was rejected.


          = 5:  The Rayleigh Quotient Iteration failed to converge to


                full accuracy in MAXITR steps.

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Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

 SLARTV applies a vector of real plane rotations to elements of the


 real vectors x and y. For i = 1,2,...,n


    ( x(i) ) := (  c(i)  s(i) ) ( x(i) )


    ( y(i) )    ( -s(i)  c(i) ) ( y(i) )

N

          N is INTEGER


          The number of plane rotations to be applied.

X

          X is REAL array,


                         dimension (1+(N-1)*INCX)


          The vector x.

INCX

          INCX is INTEGER


          The increment between elements of X. INCX > 0.

Y

          Y is REAL array,


                         dimension (1+(N-1)*INCY)


          The vector y.

INCY

          INCY is INTEGER


          The increment between elements of Y. INCY > 0.

C

          C is REAL array, dimension (1+(N-1)*INCC)


          The cosines of the plane rotations.

S

          S is REAL array, dimension (1+(N-1)*INCC)


          The sines of the plane rotations.

INCC

          INCC is INTEGER


          The increment between elements of C and S. INCC > 0.

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 SLASWP performs a series of row interchanges on the matrix A.


 One row interchange is initiated for each of rows K1 through K2 of A.

N

          N is INTEGER


          The number of columns of the matrix A.

A

          A is REAL array, dimension (LDA,N)


          On entry, the matrix of column dimension N to which the row


          interchanges will be applied.


          On exit, the permuted matrix.

LDA

          LDA is INTEGER


          The leading dimension of the array A.

K1

          K1 is INTEGER


          The first element of IPIV for which a row interchange will


          be done.

K2

          K2 is INTEGER


          (K2-K1+1) is the number of elements of IPIV for which a row


          interchange will be done.

IPIV

          IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))


          The vector of pivot indices. Only the elements in positions


          K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.


          IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be


          interchanged.

INCX

          INCX is INTEGER


          The increment between successive values of IPIV. If INCX


          is negative, the pivots are applied in reverse order.

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  Modified by


   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

 SLATBS solves one of the triangular systems


    A *x = s*b  or  A**T*x = s*b


 with scaling to prevent overflow, where A is an upper or lower


 triangular band matrix.  Here A**T denotes the transpose of A, x and b


 are n-element vectors, and s is a scaling factor, usually less than


 or equal to 1, chosen so that the components of x will be less than


 the overflow threshold.  If the unscaled problem will not cause


 overflow, the Level 2 BLAS routine STBSV is called.  If the matrix A


 is singular (A(j,j) = 0 for some j), then s is set to 0 and a


 non-trivial solution to A*x = 0 is returned.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the matrix A is upper or lower triangular.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

TRANS

          TRANS is CHARACTER*1


          Specifies the operation applied to A.


          = 'N':  Solve A * x = s*b  (No transpose)


          = 'T':  Solve A**T* x = s*b  (Transpose)


          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1


          Specifies whether or not the matrix A is unit triangular.


          = 'N':  Non-unit triangular


          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1


          Specifies whether CNORM has been set or not.


          = 'Y':  CNORM contains the column norms on entry


          = 'N':  CNORM is not set on entry.  On exit, the norms will


                  be computed and stored in CNORM.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER


          The number of subdiagonals or superdiagonals in the


          triangular matrix A.  KD >= 0.

AB

          AB is REAL array, dimension (LDAB,N)


          The upper or lower triangular band matrix A, stored in the


          first KD+1 rows of the array. The j-th column of A is stored


          in the j-th column of the array AB as follows:


          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;


          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= KD+1.

X

          X is REAL array, dimension (N)


          On entry, the right hand side b of the triangular system.


          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is REAL


          The scaling factor s for the triangular system


             A * x = s*b  or  A**T* x = s*b.


          If SCALE = 0, the matrix A is singular or badly scaled, and


          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is REAL array, dimension (N)


          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)


          contains the norm of the off-diagonal part of the j-th column


          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal


          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)


          must be greater than or equal to the 1-norm.


          If NORMIN = 'N', CNORM is an output argument and CNORM(j)


          returns the 1-norm of the offdiagonal part of the j-th column


          of A.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -k, the k-th argument had an illegal value

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  A rough bound on x is computed; if that is less than overflow, STBSV


  is called, otherwise, specific code is used which checks for possible


  overflow or divide-by-zero at every operation.


  A columnwise scheme is used for solving A*x = b.  The basic algorithm


  if A is lower triangular is


       x[1:n] := b[1:n]


       for j = 1, ..., n


            x(j) := x(j) / A(j,j)


            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]


       end


  Define bounds on the components of x after j iterations of the loop:


     M(j) = bound on x[1:j]


     G(j) = bound on x[j+1:n]


  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.


  Then for iteration j+1 we have


     M(j+1) <= G(j) / | A(j+1,j+1) |


     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |


            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )


  where CNORM(j+1) is greater than or equal to the infinity-norm of


  column j+1 of A, not counting the diagonal.  Hence


     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )


                  1<=i<=j


  and


     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )


                                   1<=i< j


  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STBSV if the


  reciprocal of the largest M(j), j=1,..,n, is larger than


  max(underflow, 1/overflow).


  The bound on x(j) is also used to determine when a step in the


  columnwise method can be performed without fear of overflow.  If


  the computed bound is greater than a large constant, x is scaled to


  prevent overflow, but if the bound overflows, x is set to 0, x(j) to


  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.


  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic


  algorithm for A upper triangular is


       for j = 1, ..., n


            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)


       end


  We simultaneously compute two bounds


       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j


       M(j) = bound on x(i), 1<=i<=j


  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we


  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.


  Then the bound on x(j) is


       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |


            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )


                      1<=i<=j


  and we can safely call STBSV if 1/M(n) and 1/G(n) are both greater


  than max(underflow, 1/overflow).

 SLATDF uses the LU factorization of the n-by-n matrix Z computed by


 SGETC2 and computes a contribution to the reciprocal Dif-estimate


 by solving Z * x = b for x, and choosing the r.h.s. b such that


 the norm of x is as large as possible. On entry RHS = b holds the


 contribution from earlier solved sub-systems, and on return RHS = x.


 The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q,


 where P and Q are permutation matrices. L is lower triangular with


 unit diagonal elements and U is upper triangular.

IJOB

          IJOB is INTEGER


          IJOB = 2: First compute an approximative null-vector e


              of Z using SGECON, e is normalized and solve for


              Zx = +-e - f with the sign giving the greater value


              of 2-norm(x). About 5 times as expensive as Default.


          IJOB .ne. 2: Local look ahead strategy where all entries of


              the r.h.s. b is chosen as either +1 or -1 (Default).

N

          N is INTEGER


          The number of columns of the matrix Z.

Z

          Z is REAL array, dimension (LDZ, N)


          On entry, the LU part of the factorization of the n-by-n


          matrix Z computed by SGETC2:  Z = P * L * U * Q

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDA >= max(1, N).

RHS

          RHS is REAL array, dimension N.


          On entry, RHS contains contributions from other subsystems.


          On exit, RHS contains the solution of the subsystem with


          entries according to the value of IJOB (see above).

RDSUM

          RDSUM is REAL


          On entry, the sum of squares of computed contributions to


          the Dif-estimate under computation by STGSYL, where the


          scaling factor RDSCAL (see below) has been factored out.


          On exit, the corresponding sum of squares updated with the


          contributions from the current sub-system.


          If TRANS = 'T' RDSUM is not touched.


          NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.

RDSCAL

          RDSCAL is REAL


          On entry, scaling factor used to prevent overflow in RDSUM.


          On exit, RDSCAL is updated w.r.t. the current contributions


          in RDSUM.


          If TRANS = 'T', RDSCAL is not touched.


          NOTE: RDSCAL only makes sense when STGSY2 is called by


                STGSYL.

IPIV

          IPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= i <= N, row i of the


          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension (N).


          The pivot indices; for 1 <= j <= N, column j of the


          matrix has been interchanged with column JPIV(j).

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This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

  [1] Bo Kagstrom and Lars Westin,


      Generalized Schur Methods with Condition Estimators for


      Solving the Generalized Sylvester Equation, IEEE Transactions


      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.


  [2] Peter Poromaa,


      On Efficient and Robust Estimators for the Separation


      between two Regular Matrix Pairs with Applications in


      Condition Estimation. Report IMINF-95.05, Departement of


      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

 SLATPS solves one of the triangular systems


    A *x = s*b  or  A**T*x = s*b


 with scaling to prevent overflow, where A is an upper or lower


 triangular matrix stored in packed form.  Here A**T denotes the


 transpose of A, x and b are n-element vectors, and s is a scaling


 factor, usually less than or equal to 1, chosen so that the


 components of x will be less than the overflow threshold.  If the


 unscaled problem will not cause overflow, the Level 2 BLAS routine


 STPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),


 then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the matrix A is upper or lower triangular.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

TRANS

          TRANS is CHARACTER*1


          Specifies the operation applied to A.


          = 'N':  Solve A * x = s*b  (No transpose)


          = 'T':  Solve A**T* x = s*b  (Transpose)


          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1


          Specifies whether or not the matrix A is unit triangular.


          = 'N':  Non-unit triangular


          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1


          Specifies whether CNORM has been set or not.


          = 'Y':  CNORM contains the column norms on entry


          = 'N':  CNORM is not set on entry.  On exit, the norms will


                  be computed and stored in CNORM.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

AP

          AP is REAL array, dimension (N*(N+1)/2)


          The upper or lower triangular matrix A, packed columnwise in


          a linear array.  The j-th column of A is stored in the array


          AP as follows:


          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;


          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

X

          X is REAL array, dimension (N)


          On entry, the right hand side b of the triangular system.


          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is REAL


          The scaling factor s for the triangular system


             A * x = s*b  or  A**T* x = s*b.


          If SCALE = 0, the matrix A is singular or badly scaled, and


          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is REAL array, dimension (N)


          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)


          contains the norm of the off-diagonal part of the j-th column


          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal


          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)


          must be greater than or equal to the 1-norm.


          If NORMIN = 'N', CNORM is an output argument and CNORM(j)


          returns the 1-norm of the offdiagonal part of the j-th column


          of A.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -k, the k-th argument had an illegal value

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  A rough bound on x is computed; if that is less than overflow, STPSV


  is called, otherwise, specific code is used which checks for possible


  overflow or divide-by-zero at every operation.


  A columnwise scheme is used for solving A*x = b.  The basic algorithm


  if A is lower triangular is


       x[1:n] := b[1:n]


       for j = 1, ..., n


            x(j) := x(j) / A(j,j)


            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]


       end


  Define bounds on the components of x after j iterations of the loop:


     M(j) = bound on x[1:j]


     G(j) = bound on x[j+1:n]


  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.


  Then for iteration j+1 we have


     M(j+1) <= G(j) / | A(j+1,j+1) |


     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |


            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )


  where CNORM(j+1) is greater than or equal to the infinity-norm of


  column j+1 of A, not counting the diagonal.  Hence


     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )


                  1<=i<=j


  and


     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )


                                   1<=i< j


  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STPSV if the


  reciprocal of the largest M(j), j=1,..,n, is larger than


  max(underflow, 1/overflow).


  The bound on x(j) is also used to determine when a step in the


  columnwise method can be performed without fear of overflow.  If


  the computed bound is greater than a large constant, x is scaled to


  prevent overflow, but if the bound overflows, x is set to 0, x(j) to


  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.


  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic


  algorithm for A upper triangular is


       for j = 1, ..., n


            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)


       end


  We simultaneously compute two bounds


       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j


       M(j) = bound on x(i), 1<=i<=j


  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we


  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.


  Then the bound on x(j) is


       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |


            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )


                      1<=i<=j


  and we can safely call STPSV if 1/M(n) and 1/G(n) are both greater


  than max(underflow, 1/overflow).

 SLATRS solves one of the triangular systems


    A *x = s*b  or  A**T*x = s*b


 with scaling to prevent overflow.  Here A is an upper or lower


 triangular matrix, A**T denotes the transpose of A, x and b are


 n-element vectors, and s is a scaling factor, usually less than


 or equal to 1, chosen so that the components of x will be less than


 the overflow threshold.  If the unscaled problem will not cause


 overflow, the Level 2 BLAS routine STRSV is called.  If the matrix A


 is singular (A(j,j) = 0 for some j), then s is set to 0 and a


 non-trivial solution to A*x = 0 is returned.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the matrix A is upper or lower triangular.


          = 'U':  Upper triangular


          = 'L':  Lower triangular

TRANS

          TRANS is CHARACTER*1


          Specifies the operation applied to A.


          = 'N':  Solve A * x = s*b  (No transpose)


          = 'T':  Solve A**T* x = s*b  (Transpose)


          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1


          Specifies whether or not the matrix A is unit triangular.


          = 'N':  Non-unit triangular


          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1


          Specifies whether CNORM has been set or not.


          = 'Y':  CNORM contains the column norms on entry


          = 'N':  CNORM is not set on entry.  On exit, the norms will


                  be computed and stored in CNORM.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          The triangular matrix A.  If UPLO = 'U', the leading n by n


          upper triangular part of the array A contains the upper


          triangular matrix, and the strictly lower triangular part of


          A is not referenced.  If UPLO = 'L', the leading n by n lower


          triangular part of the array A contains the lower triangular


          matrix, and the strictly upper triangular part of A is not


          referenced.  If DIAG = 'U', the diagonal elements of A are


          also not referenced and are assumed to be 1.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max (1,N).

X

          X is REAL array, dimension (N)


          On entry, the right hand side b of the triangular system.


          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is REAL


          The scaling factor s for the triangular system


             A * x = s*b  or  A**T* x = s*b.


          If SCALE = 0, the matrix A is singular or badly scaled, and


          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is REAL array, dimension (N)


          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)


          contains the norm of the off-diagonal part of the j-th column


          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal


          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)


          must be greater than or equal to the 1-norm.


          If NORMIN = 'N', CNORM is an output argument and CNORM(j)


          returns the 1-norm of the offdiagonal part of the j-th column


          of A.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -k, the k-th argument had an illegal value

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  A rough bound on x is computed; if that is less than overflow, STRSV


  is called, otherwise, specific code is used which checks for possible


  overflow or divide-by-zero at every operation.


  A columnwise scheme is used for solving A*x = b.  The basic algorithm


  if A is lower triangular is


       x[1:n] := b[1:n]


       for j = 1, ..., n


            x(j) := x(j) / A(j,j)


            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]


       end


  Define bounds on the components of x after j iterations of the loop:


     M(j) = bound on x[1:j]


     G(j) = bound on x[j+1:n]


  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.


  Then for iteration j+1 we have


     M(j+1) <= G(j) / | A(j+1,j+1) |


     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |


            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )


  where CNORM(j+1) is greater than or equal to the infinity-norm of


  column j+1 of A, not counting the diagonal.  Hence


     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )


                  1<=i<=j


  and


     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )


                                   1<=i< j


  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the


  reciprocal of the largest M(j), j=1,..,n, is larger than


  max(underflow, 1/overflow).


  The bound on x(j) is also used to determine when a step in the


  columnwise method can be performed without fear of overflow.  If


  the computed bound is greater than a large constant, x is scaled to


  prevent overflow, but if the bound overflows, x is set to 0, x(j) to


  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.


  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic


  algorithm for A upper triangular is


       for j = 1, ..., n


            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)


       end


  We simultaneously compute two bounds


       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j


       M(j) = bound on x(i), 1<=i<=j


  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we


  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.


  Then the bound on x(j) is


       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |


            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )


                      1<=i<=j


  and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater


  than max(underflow, 1/overflow).

 SLAUU2 computes the product U * U**T or L**T * L, where the triangular


 factor U or L is stored in the upper or lower triangular part of


 the array A.


 If UPLO = 'U' or 'u' then the upper triangle of the result is stored,


 overwriting the factor U in A.


 If UPLO = 'L' or 'l' then the lower triangle of the result is stored,


 overwriting the factor L in A.


 This is the unblocked form of the algorithm, calling Level 2 BLAS.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the triangular factor stored in the array A


          is upper or lower triangular:


          = 'U':  Upper triangular


          = 'L':  Lower triangular

N

          N is INTEGER


          The order of the triangular factor U or L.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the triangular factor U or L.


          On exit, if UPLO = 'U', the upper triangle of A is


          overwritten with the upper triangle of the product U * U**T;


          if UPLO = 'L', the lower triangle of A is overwritten with


          the lower triangle of the product L**T * L.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -k, the k-th argument had an illegal value

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 SLAUUM computes the product U * U**T or L**T * L, where the triangular


 factor U or L is stored in the upper or lower triangular part of


 the array A.


 If UPLO = 'U' or 'u' then the upper triangle of the result is stored,


 overwriting the factor U in A.


 If UPLO = 'L' or 'l' then the lower triangle of the result is stored,


 overwriting the factor L in A.


 This is the blocked form of the algorithm, calling Level 3 BLAS.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the triangular factor stored in the array A


          is upper or lower triangular:


          = 'U':  Upper triangular


          = 'L':  Lower triangular

N

          N is INTEGER


          The order of the triangular factor U or L.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the triangular factor U or L.


          On exit, if UPLO = 'U', the upper triangle of A is


          overwritten with the upper triangle of the product U * U**T;


          if UPLO = 'L', the lower triangle of A is overwritten with


          the lower triangle of the product L**T * L.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -k, the k-th argument had an illegal value

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 SRSCL multiplies an n-element real vector x by the real scalar 1/a.


 This is done without overflow or underflow as long as


 the final result x/a does not overflow or underflow.

N

          N is INTEGER


          The number of components of the vector x.

SA

          SA is REAL


          The scalar a which is used to divide each component of x.


          SA must be >= 0, or the subroutine will divide by zero.

SX

          SX is REAL array, dimension


                         (1+(N-1)*abs(INCX))


          The n-element vector x.

INCX

          INCX is INTEGER


          The increment between successive values of the vector SX.


          > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n

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 STPRFB applies a real 'triangular-pentagonal' block reflector H or its


 transpose H**T to a real matrix C, which is composed of two


 blocks A and B, either from the left or right.

SIDE

          SIDE is CHARACTER*1


          = 'L': apply H or H**T from the Left


          = 'R': apply H or H**T from the Right

TRANS

          TRANS is CHARACTER*1


          = 'N': apply H (No transpose)


          = 'T': apply H**T (Transpose)

DIRECT

          DIRECT is CHARACTER*1


          Indicates how H is formed from a product of elementary


          reflectors


          = 'F': H = H(1) H(2) . . . H(k) (Forward)


          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1


          Indicates how the vectors which define the elementary


          reflectors are stored:


          = 'C': Columns


          = 'R': Rows

M

          M is INTEGER


          The number of rows of the matrix B.


          M >= 0.

N

          N is INTEGER


          The number of columns of the matrix B.


          N >= 0.

K

          K is INTEGER


          The order of the matrix T, i.e. the number of elementary


          reflectors whose product defines the block reflector.


          K >= 0.

L

          L is INTEGER


          The order of the trapezoidal part of V.


          K >= L >= 0.  See Further Details.

V

          V is REAL array, dimension


                                (LDV,K) if STOREV = 'C'


                                (LDV,M) if STOREV = 'R' and SIDE = 'L'


                                (LDV,N) if STOREV = 'R' and SIDE = 'R'


          The pentagonal matrix V, which contains the elementary reflectors


          H(1), H(2), ..., H(K).  See Further Details.

LDV

          LDV is INTEGER


          The leading dimension of the array V.


          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);


          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);


          if STOREV = 'R', LDV >= K.

T

          T is REAL array, dimension (LDT,K)


          The triangular K-by-K matrix T in the representation of the


          block reflector.

LDT

          LDT is INTEGER


          The leading dimension of the array T.


          LDT >= K.

A

          A is REAL array, dimension


          (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'


          On entry, the K-by-N or M-by-K matrix A.


          On exit, A is overwritten by the corresponding block of


          H*C or H**T*C or C*H or C*H**T.  See Further Details.

LDA

          LDA is INTEGER


          The leading dimension of the array A.


          If SIDE = 'L', LDA >= max(1,K);


          If SIDE = 'R', LDA >= max(1,M).

B

          B is REAL array, dimension (LDB,N)


          On entry, the M-by-N matrix B.


          On exit, B is overwritten by the corresponding block of


          H*C or H**T*C or C*H or C*H**T.  See Further Details.

LDB

          LDB is INTEGER


          The leading dimension of the array B.


          LDB >= max(1,M).

WORK

          WORK is REAL array, dimension


          (LDWORK,N) if SIDE = 'L',


          (LDWORK,K) if SIDE = 'R'.

LDWORK

          LDWORK is INTEGER


          The leading dimension of the array WORK.


          If SIDE = 'L', LDWORK >= K;


          if SIDE = 'R', LDWORK >= M.

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  The matrix C is a composite matrix formed from blocks A and B.


  The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,


  and if SIDE = 'L', A is of size K-by-N.


  If SIDE = 'R' and DIRECT = 'F', C = [A B].


  If SIDE = 'L' and DIRECT = 'F', C = [A]


                                      [B].


  If SIDE = 'R' and DIRECT = 'B', C = [B A].


  If SIDE = 'L' and DIRECT = 'B', C = [B]


                                      [A].


  The pentagonal matrix V is composed of a rectangular block V1 and a


  trapezoidal block V2.  The size of the trapezoidal block is determined by


  the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;


  if L=0, there is no trapezoidal block, thus V = V1 is rectangular.


  If DIRECT = 'F' and STOREV = 'C':  V = [V1]


                                         [V2]


     - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)


  If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]


     - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)


  If DIRECT = 'B' and STOREV = 'C':  V = [V2]


                                         [V1]


     - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)


  If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]


     - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)


  If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.


  If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.


  If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.


  If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.