dtgsyl.f (3) Linux Manual Page

dtgsyl.f –

Synopsis

Functions/Subroutines

subroutine dtgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
DTGSYL

Function/Subroutine Documentation

subroutine dtgsyl (characterTRANS, integerIJOB, integerM, integerN, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( ldc, * )C, integerLDC, double precision, dimension( ldd, * )D, integerLDD, double precision, dimension( lde, * )E, integerLDE, double precision, dimension( ldf, * )F, integerLDF, double precisionSCALE, double precisionDIF, double precision, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)

DTGSYL

Purpose:

 DTGSYL solves the generalized Sylvester equation:

 A * R – L * B = scale * C (1)

 D * R – L * E = scale * F

 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 (real) 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 (1) is equivalent to solve Zx = scale b, where

 Z is defined as

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

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

 Here 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.

 If TRANS = ‘T’, DTGSYL solves the transposed system Z**T*y = scale*b,

 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 (TRANS = ‘T’) is used to compute an one-norm-based estimate

 of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)

 and (B,E), using DLACON.

 If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate

 of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the

 reciprocal of the smallest singular value of Z. See [1-2] for more

 information.

 This is a level 3 BLAS algorithm.

 

Parameters:

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: The functionality of 0 and 3.

 =2: The functionality of 0 and 4.

 =3: Only an estimate of Dif[(A,D), (B,E)] is computed.

 (look ahead strategy IJOB = 1 is used).

 =4: Only an estimate of Dif[(A,D), (B,E)] is computed.

 ( DGECON on sub-systems is used ).

 Not referenced if TRANS = ‘T’.

M

 M is INTEGER

 The order of the matrices A and D, and the row dimension of

 the matrices C, F, R and L.

N

 N is INTEGER

 The order of the matrices B and E, and the column dimension

 of the matrices C, F, R and L.

A

 A is DOUBLE PRECISION array, dimension (LDA, M)

 The upper quasi triangular matrix A.

LDA

 LDA is INTEGER

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

B

 B is DOUBLE PRECISION array, dimension (LDB, N)

 The upper quasi triangular matrix B.

LDB

 LDB is INTEGER

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

C

 C is DOUBLE PRECISION array, dimension (LDC, N)

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

 equation in (1) or (3).

 On exit, if IJOB = 0, 1 or 2, C has been overwritten by

 the solution R. If IJOB = 3 or 4 and TRANS = ‘N’, C holds R,

 the solution achieved during the computation of the

 Dif-estimate.

LDC

 LDC is INTEGER

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

D

 D is DOUBLE PRECISION array, dimension (LDD, M)

 The upper triangular matrix D.

LDD

 LDD is INTEGER

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

E

 E is DOUBLE PRECISION array, dimension (LDE, N)

 The upper triangular matrix E.

LDE

 LDE is INTEGER

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

F

 F is DOUBLE PRECISION array, dimension (LDF, N)

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

 equation in (1) or (3).

 On exit, if IJOB = 0, 1 or 2, F has been overwritten by

 the solution L. If IJOB = 3 or 4 and TRANS = ‘N’, F holds L,

 the solution achieved during the computation of the

 Dif-estimate.

LDF

 LDF is INTEGER

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

DIF

 DIF is DOUBLE PRECISION

 On exit DIF is the reciprocal of a lower bound of the

 reciprocal of the Dif-function, i.e. DIF is an upper bound of

 Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).

 IF IJOB = 0 or TRANS = ‘T’, DIF is not touched.

SCALE

 SCALE is DOUBLE PRECISION

 On exit SCALE is the scaling factor in (1) or (3).

 If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,

 to a slightly perturbed system but the input matrices A, B, D

 and E have not been changed. If SCALE = 0, C and F hold the

 solutions R and L, respectively, to the homogeneous system

 with C = F = 0. Normally, SCALE = 1.

WORK

 WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))

 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

 LWORK is INTEGER

 The dimension of the array WORK. LWORK > = 1.

 If IJOB = 1 or 2 and TRANS = ‘N’, LWORK >= max(1,2*M*N).

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

 only calculates the optimal size of the WORK array, returns

 this value as the first entry of the WORK array, and no error

 message related to LWORK is issued by XERBLA.

IWORK

 IWORK is INTEGER array, dimension (M+N+6)

INFO

 INFO is INTEGER

 =0: successful exit

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

 >0: (A, D) and (B, E) have common or close eigenvalues.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Contributors:

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

References:

 [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

 for Solving the Generalized Sylvester Equation and Estimating the

 Separation between Regular Matrix Pairs, Report UMINF – 93.23,

 Department of Computing Science, Umea University, S-901 87 Umea,

 Sweden, December 1993, Revised April 1994, Also as LAPACK Working

 Note 75. To appear in ACM Trans. on Math. Software, Vol 22,

 No 1, 1996.

 [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester

 Equation (AR – LB, DR – LE ) = (C, F), SIAM J. Matrix Anal.

 Appl., H(4):1045-1060, 1994

 [3] B. Kagstrom and L. 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.

 

Definition at line 298 of file dtgsyl.f.

Author

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