CLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix


    ( ( A, B );( B, C ) )


 provided the norm of the matrix of eigenvectors is larger than


 some threshold value.


 RT1 is the eigenvalue of larger absolute value, and RT2 of


 smaller absolute value.  If the eigenvectors are computed, then


 on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence


 [  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]


 [ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]

A

          A is COMPLEX


          The ( 1, 1 ) element of input matrix.

B

          B is COMPLEX


          The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element


          is also given by B, since the 2-by-2 matrix is symmetric.

C

          C is COMPLEX


          The ( 2, 2 ) element of input matrix.

RT1

          RT1 is COMPLEX


          The eigenvalue of larger modulus.

RT2

          RT2 is COMPLEX


          The eigenvalue of smaller modulus.

EVSCAL

          EVSCAL is COMPLEX


          The complex value by which the eigenvector matrix was scaled


          to make it orthonormal.  If EVSCAL is zero, the eigenvectors


          were not computed.  This means one of two things:  the 2-by-2


          matrix could not be diagonalized, or the norm of the matrix


          of eigenvectors before scaling was larger than the threshold


          value THRESH (set below).

CS1

          CS1 is COMPLEX

SN1

          SN1 is COMPLEX


          If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector


          for RT1.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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


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


 complex symmetric matrix A.

CLANSY

    CLANSY = ( 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 CLANSY as described


          above.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the upper or lower triangular part of the


          symmetric matrix A is to be referenced.


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


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

N

          N is INTEGER


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


          set to zero.

A

          A is COMPLEX array, dimension (LDA,N)


          The symmetric matrix A.  If UPLO = 'U', the leading n by n


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading n by n lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A 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' or '1' or 'O'; otherwise,


          WORK is not referenced.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CLAQSY 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.

A

          A is COMPLEX array, dimension (LDA,N)


          On entry, the symmetric matrix A.  If UPLO = 'U', the leading


          n by n upper triangular part of A contains the upper


          triangular part of the matrix A, and the strictly lower


          triangular part of A is not referenced.  If UPLO = 'L', the


          leading n by n lower triangular part of A contains the lower


          triangular part of the matrix A, and the strictly upper


          triangular part of A is not referenced.


          On exit, if EQUED = 'Y', the equilibrated matrix:


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

LDA

          LDA is INTEGER


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CSYMV  performs the matrix-vector  operation


    y := alpha*A*x + beta*y,


 where alpha and beta are scalars, x and y are n element vectors and


 A is an n by n symmetric matrix.

UPLO

          UPLO is CHARACTER*1


           On entry, UPLO specifies whether the upper or lower


           triangular part of the array A is to be referenced as


           follows:


              UPLO = 'U' or 'u'   Only the upper triangular part of A


                                  is to be referenced.


              UPLO = 'L' or 'l'   Only the lower triangular part of A


                                  is to be referenced.


           Unchanged on exit.

N

          N is INTEGER


           On entry, N specifies the order of the matrix A.


           N must be at least zero.


           Unchanged on exit.

ALPHA

          ALPHA is COMPLEX


           On entry, ALPHA specifies the scalar alpha.


           Unchanged on exit.

A

          A is COMPLEX array, dimension ( LDA, N )


           Before entry, with  UPLO = 'U' or 'u', the leading n by n


           upper triangular part of the array A must contain the upper


           triangular part of the symmetric matrix and the strictly


           lower triangular part of A is not referenced.


           Before entry, with UPLO = 'L' or 'l', the leading n by n


           lower triangular part of the array A must contain the lower


           triangular part of the symmetric matrix and the strictly


           upper triangular part of A is not referenced.


           Unchanged on exit.

LDA

          LDA is INTEGER


           On entry, LDA specifies the first dimension of A as declared


           in the calling (sub) program. LDA must be at least


           max( 1, N ).


           Unchanged on exit.

X

          X is COMPLEX array, dimension at least


           ( 1 + ( N - 1 )*abs( INCX ) ).


           Before entry, the incremented array X must contain the N-


           element vector x.


           Unchanged on exit.

INCX

          INCX is INTEGER


           On entry, INCX specifies the increment for the elements of


           X. INCX must not be zero.


           Unchanged on exit.

BETA

          BETA is COMPLEX


           On entry, BETA specifies the scalar beta. When BETA is


           supplied as zero then Y need not be set on input.


           Unchanged on exit.

Y

          Y is COMPLEX array, dimension at least


           ( 1 + ( N - 1 )*abs( INCY ) ).


           Before entry, the incremented array Y must contain the n


           element vector y. On exit, Y is overwritten by the updated


           vector y.

INCY

          INCY is INTEGER


           On entry, INCY specifies the increment for the elements of


           Y. INCY must not be zero.


           Unchanged on exit.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CSYR   performs the symmetric rank 1 operation


    A := alpha*x*x**H + A,


 where alpha is a complex scalar, x is an n element vector and A is an


 n by n symmetric matrix.

UPLO

          UPLO is CHARACTER*1


           On entry, UPLO specifies whether the upper or lower


           triangular part of the array A is to be referenced as


           follows:


              UPLO = 'U' or 'u'   Only the upper triangular part of A


                                  is to be referenced.


              UPLO = 'L' or 'l'   Only the lower triangular part of A


                                  is to be referenced.


           Unchanged on exit.

N

          N is INTEGER


           On entry, N specifies the order of the matrix A.


           N must be at least zero.


           Unchanged on exit.

ALPHA

          ALPHA is COMPLEX


           On entry, ALPHA specifies the scalar alpha.


           Unchanged on exit.

X

          X is COMPLEX array, dimension at least


           ( 1 + ( N - 1 )*abs( INCX ) ).


           Before entry, the incremented array X must contain the N-


           element vector x.


           Unchanged on exit.

INCX

          INCX is INTEGER


           On entry, INCX specifies the increment for the elements of


           X. INCX must not be zero.


           Unchanged on exit.

A

          A is COMPLEX array, dimension ( LDA, N )


           Before entry, with  UPLO = 'U' or 'u', the leading n by n


           upper triangular part of the array A must contain the upper


           triangular part of the symmetric matrix and the strictly


           lower triangular part of A is not referenced. On exit, the


           upper triangular part of the array A is overwritten by the


           upper triangular part of the updated matrix.


           Before entry, with UPLO = 'L' or 'l', the leading n by n


           lower triangular part of the array A must contain the lower


           triangular part of the symmetric matrix and the strictly


           upper triangular part of A is not referenced. On exit, the


           lower triangular part of the array A is overwritten by the


           lower triangular part of the updated matrix.

LDA

          LDA is INTEGER


           On entry, LDA specifies the first dimension of A as declared


           in the calling (sub) program. LDA must be at least


           max( 1, N ).


           Unchanged on exit.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CSYSWAPR applies an elementary permutation on the rows and the columns of


 a symmetric matrix.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the details of the factorization are stored


          as an upper or lower triangular matrix.


          = 'U':  Upper triangular, form is A = U*D*U**T;


          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA,N)


          On entry, the N-by-N matrix A. On exit, the permuted matrix


          where the rows I1 and I2 and columns I1 and I2 are interchanged.


          If UPLO = 'U', the interchanges are applied to the upper


          triangular part and the strictly lower triangular part of A is


          not referenced; if UPLO = 'L', the interchanges are applied to


          the lower triangular part and the part of A above the diagonal


          is not referenced.

LDA

          LDA is INTEGER


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

I1

          I1 is INTEGER


          Index of the first row to swap

I2

          I2 is INTEGER


          Index of the second row to swap

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CTGSY2 solves the generalized Sylvester equation


             A * R - L * B = scale *  C               (1)


             D * R - L * E = scale * F


 using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,


 (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,


 N-by-N and M-by-N, respectively. A, B, D and E are upper triangular


 (i.e., (A,D) and (B,E) in generalized Schur form).


 The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output


 scaling factor chosen to avoid overflow.


 In matrix notation solving equation (1) corresponds to solve


 Zx = scale * b, where Z is defined as


        Z = [ kron(In, A)  -kron(B**H, Im) ]             (2)


            [ kron(In, D)  -kron(E**H, Im) ],


 Ik is the identity matrix of size k and X**H is the transpose of X.


 kron(X, Y) is the Kronecker product between the matrices X and Y.


 If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b


 is solved for, which is equivalent to solve for R and L in


             A**H * R  + D**H * L   = scale * C           (3)


             R  * B**H + L  * E**H  = scale * -F


 This case is used to compute an estimate of Dif[(A, D), (B, E)] =


 = sigma_min(Z) using reverse communication with CLACON.


 CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL


 of an upper bound on the separation between to matrix pairs. Then


 the input (A, D), (B, E) are sub-pencils of two matrix pairs in


 CTGSYL.

TRANS

          TRANS is CHARACTER*1


          = 'N': solve the generalized Sylvester equation (1).


          = 'T': solve the 'transposed' system (3).

IJOB

          IJOB is INTEGER


          Specifies what kind of functionality to be performed.


          = 0: solve (1) only.


          = 1: A contribution from this subsystem to a Frobenius


               norm-based estimate of the separation between two matrix


               pairs is computed. (look ahead strategy is used).


          = 2: A contribution from this subsystem to a Frobenius


               norm-based estimate of the separation between two matrix


               pairs is computed. (SGECON on sub-systems is used.)


          Not referenced if TRANS = 'T'.

M

          M is INTEGER


          On entry, M specifies the order of A and D, and the row


          dimension of C, F, R and L.

N

          N is INTEGER


          On entry, N specifies the order of B and E, and the column


          dimension of C, F, R and L.

A

          A is COMPLEX array, dimension (LDA, M)


          On entry, A contains an upper triangular matrix.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, B contains an upper triangular matrix.

LDB

          LDB is INTEGER


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

C

          C is COMPLEX array, dimension (LDC, N)


          On entry, C contains the right-hand-side of the first matrix


          equation in (1).


          On exit, if IJOB = 0, C has been overwritten by the solution


          R.

LDC

          LDC is INTEGER


          The leading dimension of the matrix C. LDC >= max(1, M).

D

          D is COMPLEX array, dimension (LDD, M)


          On entry, D contains an upper triangular matrix.

LDD

          LDD is INTEGER


          The leading dimension of the matrix D. LDD >= max(1, M).

E

          E is COMPLEX array, dimension (LDE, N)


          On entry, E contains an upper triangular matrix.

LDE

          LDE is INTEGER


          The leading dimension of the matrix E. LDE >= max(1, N).

F

          F is COMPLEX array, dimension (LDF, N)


          On entry, F contains the right-hand-side of the second matrix


          equation in (1).


          On exit, if IJOB = 0, F has been overwritten by the solution


          L.

LDF

          LDF is INTEGER


          The leading dimension of the matrix F. LDF >= max(1, M).

SCALE

          SCALE is REAL


          On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions


          R and L (C and F on entry) will hold the solutions to a


          slightly perturbed system but the input matrices A, B, D and


          E have not been changed. If SCALE = 0, R and L will hold the


          solutions to the homogeneous system with C = F = 0.


          Normally, SCALE = 1.

RDSUM

          RDSUM is REAL


          On entry, the sum of squares of computed contributions to


          the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by


          CTGSYL.

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 CTGSY2 is called by


          CTGSYL.

INFO

          INFO is INTEGER


          On exit, if INFO is set to


            =0: Successful exit


            <0: If INFO = -i, input argument number i is illegal.


            >0: The matrix pairs (A, D) and (B, E) have common or very


                close eigenvalues.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.