SLANSY  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.

SLANSY

    SLANSY = ( 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 SLANSY 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, SLANSY is


          set to zero.

A

          A is REAL 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.

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 SLAQSY 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 REAL 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.

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 SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in


        op(TL)*X + ISGN*X*op(TR) = SCALE*B,


 where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or


 -1.  op(T) = T or T**T, where T**T denotes the transpose of T.

LTRANL

          LTRANL is LOGICAL


          On entry, LTRANL specifies the op(TL):


             = .FALSE., op(TL) = TL,


             = .TRUE., op(TL) = TL**T.

LTRANR

          LTRANR is LOGICAL


          On entry, LTRANR specifies the op(TR):


            = .FALSE., op(TR) = TR,


            = .TRUE., op(TR) = TR**T.

ISGN

          ISGN is INTEGER


          On entry, ISGN specifies the sign of the equation


          as described before. ISGN may only be 1 or -1.

N1

          N1 is INTEGER


          On entry, N1 specifies the order of matrix TL.


          N1 may only be 0, 1 or 2.

N2

          N2 is INTEGER


          On entry, N2 specifies the order of matrix TR.


          N2 may only be 0, 1 or 2.

TL

          TL is REAL array, dimension (LDTL,2)


          On entry, TL contains an N1 by N1 matrix.

LDTL

          LDTL is INTEGER


          The leading dimension of the matrix TL. LDTL >= max(1,N1).

TR

          TR is REAL array, dimension (LDTR,2)


          On entry, TR contains an N2 by N2 matrix.

LDTR

          LDTR is INTEGER


          The leading dimension of the matrix TR. LDTR >= max(1,N2).

B

          B is REAL array, dimension (LDB,2)


          On entry, the N1 by N2 matrix B contains the right-hand


          side of the equation.

LDB

          LDB is INTEGER


          The leading dimension of the matrix B. LDB >= max(1,N1).

SCALE

          SCALE is REAL


          On exit, SCALE contains the scale factor. SCALE is chosen


          less than or equal to 1 to prevent the solution overflowing.

X

          X is REAL array, dimension (LDX,2)


          On exit, X contains the N1 by N2 solution.

LDX

          LDX is INTEGER


          The leading dimension of the matrix X. LDX >= max(1,N1).

XNORM

          XNORM is REAL


          On exit, XNORM is the infinity-norm of the solution.

INFO

          INFO is INTEGER


          On exit, INFO is set to


             0: successful exit.


             1: TL and TR have too close eigenvalues, so TL or


                TR is perturbed to get a nonsingular equation.


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


                check the inputs for errors.

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 SSYSWAPR 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 REAL array, dimension (LDA,*)


          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

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 STGSY2 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, with real entries. (A, D) and (B, E)


 must be in generalized Schur canonical form, i.e. A, B are upper


 quasi triangular and D, E are upper triangular. 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


 Z*x = scale*b, where Z is defined as


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


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


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


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


 In the process of solving (1), we solve a number of such systems


 where Dim(In), Dim(In) = 1 or 2.


 If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,


 which is equivalent to solve for R and L in


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


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


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


 sigma_min(Z) using reverse communication with SLACON.


 STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL


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


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


 STGSYL. See STGSYL for details.

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 REAL array, dimension (LDA, M)


          On entry, A contains an upper quasi triangular matrix.

LDA

          LDA is INTEGER


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

B

          B is REAL array, dimension (LDB, N)


          On entry, B contains an upper quasi triangular matrix.

LDB

          LDB is INTEGER


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

C

          C is REAL 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 REAL 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 REAL 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 REAL 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 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.

IWORK

          IWORK is INTEGER array, dimension (M+N+2)

PQ

          PQ is INTEGER


          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and


          8-by-8) solved by this routine.

INFO

          INFO is INTEGER


          On exit, if INFO is set to


            =0: Successful exit


            <0: If INFO = -i, the i-th argument had an illegal value.


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


                close eigenvalues.

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