Purpose
To reduce a real matrix pair (A,B) in generalized real Schur form to a block-diagonal form using well-conditioned non-orthogonal equivalence transformations. The condition numbers of the left and right transformations used for the reduction are roughly bounded by PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in the given matrices X and Y. The generalized Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same pair of blocks.Specification
SUBROUTINE MB04RD( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X, $ LDX, Y, LDY, NBLCKS, BLSIZE, ALPHAR, ALPHAI, $ BETA, TOL, IWORK, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER JOBX, JOBY, SORT INTEGER INFO, LDA, LDB, LDWORK, LDX, LDY, N, NBLCKS DOUBLE PRECISION PMAX, TOL C .. Array Arguments .. INTEGER BLSIZE(*), IWORK(*) DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*), $ BETA(*), DWORK(*), X(LDX,*), Y(LDY,*)Arguments
Mode Parameters
JOBX CHARACTER*1 Specifies whether or not the left transformations are accumulated, as follows: = 'N': The left transformations are not accumulated; = 'U': The left transformations are accumulated in X (the given matrix X is updated). JOBY CHARACTER*1 Specifies whether or not the right transformations are accumulated, as follows: = 'N': The right transformations are not accumulated; = 'U': The right transformations are accumulated in Y (the given matrix Y is updated). SORT CHARACTER*1 Specifies whether or not the diagonal blocks of the generalized real Schur form are reordered, as follows: = 'N': The diagonal blocks are not reordered; = 'S': The diagonal blocks are reordered before each step of reduction, so that clustered eigenvalues appear in the same pair of blocks. = 'C': The diagonal blocks are not reordered, but the "closest-neighbour" strategy is used instead of the standard "closest to the mean" strategy (see METHOD); = 'B': The diagonal blocks are reordered before each step of reduction, and the "closest-neighbour" strategy is used (see METHOD).Input/Output Parameters
N (input) INTEGER The order of the matrices A, B, X and Y. N >= 0. PMAX (input) DOUBLE PRECISION An upper bound for the absolute value of the elements of the individual transformations used for reduction (see METHOD and FURTHER COMMENTS). PMAX >= 1.0D0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N upper quasi-triangular part of this array must contain the upper quasi-triangular matrix A in the generalized real Schur form, as returned by the LAPACK Library routine DGGES. The lower triangular part below the Schur matrix is used as workspace. On exit, the leading N-by-N upper quasi-triangular part of this array contains the computed block-diagonal matrix, in real Schur canonical form, corresponding to the given matrix A. The remaining part is set to zero. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,N) On entry, the leading N-by-N upper triangular part of this array must contain the upper triangular matrix B in the generalized real Schur form, as returned by the LAPACK Library routine DGGES. The diagonal elements of B are non-negative. The strictly lower triangular part is used as workspace. The matrix B is assumed nonzero. On exit, the leading N-by-N upper triangular part of this array contains the computed upper triangular block- diagonal matrix, corresponding to the given matrix B. The remaining part is set to zero. The diagonal elements of B are non-negative. LDB INTEGER The leading dimension of the array B. LDB >= MAX(1,N). X (input/output) DOUBLE PRECISION array, dimension (LDX,*) On entry, if JOBX = 'U', the leading N-by-N part of this array must contain a given matrix X, for instance the left transformation matrix VSL returned by the LAPACK Library routine DGGES. On exit, if JOBX = 'U', the leading N-by-N part of this array contains the product of the given matrix X and the left transformation matrix that reduced (A,B) to block- diagonal form. The local transformation matrix is itself a product of non-orthogonal equivalence transformations having elements with magnitude less than or equal to PMAX. If JOBX = 'N', this array is not referenced. LDX INTEGER The leading dimension of the array X. LDX >= 1, if JOBX = 'N'; LDX >= MAX(1,N), if JOBX = 'U'. Y (input/output) DOUBLE PRECISION array, dimension (LDY,*) On entry, if JOBY = 'U', the leading N-by-N part of this array must contain a given matrix Y, for instance the right transformation matrix VSR returned by the LAPACK Library routine DGGES. On exit, if JOBY = 'U', the leading N-by-N part of this array contains the product of the given matrix Y and the right transformation matrix that reduced (A,B) to block- diagonal form. The local transformation matrix is itself a product of non-orthogonal equivalence transformations having elements with magnitude less than or equal to PMAX. If JOBY = 'N', this array is not referenced. LDY INTEGER The leading dimension of the array Y. LDY >= 1, if JOBY = 'N'; LDY >= MAX(1,N), if JOBY = 'U'. NBLCKS (output) INTEGER The number of diagonal blocks of the matrices A and B. BLSIZE (output) INTEGER array, dimension (N) The first NBLCKS elements of this array contain the orders of the resulting diagonal blocks of the matrices A and B. ALPHAR (output) DOUBLE PRECISION array, dimension (N) ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE PRECISION array, dimension (N) On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j = 1, ..., N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1, ..., N, are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative. All BETA(j) are non-negative real numbers. The quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. If A and B are obtained from general matrices using DGGES, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).Tolerances
TOL DOUBLE PRECISION If SORT = 'S' or 'B', the tolerance to be used in the ordering of the diagonal blocks of the upper triangular matrix pair. If the user sets TOL > 0, then the given value of TOL is used as an absolute tolerance: a pair of blocks i and a temporarily fixed pair of blocks 1 (the first pair of blocks of the current trailing pair of submatrices to be reduced) are considered to belong to the same cluster if their eigenvalues satisfy the following "distance" condition | lambda_1 - lambda_i | <= TOL. If the user sets TOL < 0, then the given value of TOL is used as a relative tolerance: a pair of blocks i and a temporarily fixed pair of blocks 1 are considered to belong to the same cluster if their eigenvalues satisfy, for finite lambda_j, j = 1, ..., N, | lambda_1 - lambda_i | <= | TOL | * max | lambda_j |. If the user sets TOL = 0, then an implicitly computed, default tolerance, defined by TOL = SQRT( SQRT( EPS ) ) is used instead, as a relative tolerance, where EPS is the machine precision (see LAPACK Library routine DLAMCH). The approximate symmetric chordal metric is used as "distance" of two complex, possibly infinite numbers, x and y. This metric is given by the formula d(x,y) = min( |x-y|, |1/x-1/y| ), taking into account the special cases of infinite or NaN values. If SORT = 'N' or 'C', this parameter is not referenced.Workspace
IWORK INTEGER array, dimension (N+6) DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) contains the optimal value of LDWORK. On exit, if INFO = -21, DWORK(1) returns the minimum value of LDWORK. When LDWORK = 0 is set on entry, the routine will return this value for INFO, and also set DWORK(1), but no error message related to LDWORK is issued by XERBLA. LDWORK INTEGER The dimension of the array DWORK. LDWORK >= 1, if N <= 1; LDWORK >= 4*N + 16, if N > 1. If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the DWORK array, returns this value as the first entry of the DWORK array, and no error message related to LDWORK is issued by XERBLA.Error Indicator
INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the matrix pencil defined by A and B is singular.Method
Consider first that SORT = 'N'. Let ( A A ) ( B B ) ( 11 12 ) ( 11 12 ) A = ( ), B = ( ), ( 0 A ) ( 0 B ) ( 22 ) ( 22 ) be the given matrix pair in generalized real Schur form, where initially A and B are the first pair of diagonal blocks of 11 11 dimension 1-by-1 or 2-by-2. An attempt is made to compute the transformation matrices X and Y of the form ( I V ) ( I W ) X = ( ), Y = ( ) (1) ( 0 I ) ( 0 I ) (partitioned as A and B ), so that (' denotes the transpose) ( A 0 ) ( B 0 ) ( 11 ) ( 11 ) X' A Y = ( ), X' B Y = ( ), ( 0 A ) ( 0 B ) ( 22 ) ( 22 ) and the elements of V and W do not exceed the value PMAX in magnitude. An adaptation of the standard method for solving generalized Sylvester equations [1], which controls the magnitude of the individual elements of the computed solution [2], is used to obtain V and W. When this attempt fails, a 1-by-1 (or 2-by-2) pair of diagonal blocks of (A , B ), whose eigenvalue(s) is 22 22 (are) the closest to the mean of those of (A , B ) is selected, 11 11 and moved by orthogonal equivalence transformations in the leading position of (A , B ); the moved diagonal blocks in A and B are 22 22 then added to the blocks A and B , respectively, increasing 11 11 their order by 1 (or 2). Another attempt is made to compute suitable transformation matrices X and Y with the new definitions of the blocks A , A , B , and B . After successful 11 22 11 22 transformation matrices X and Y have been obtained, they postmultiply the current transformation matrices (if JOBX = 'U' and/or JOBY = 'U') and the whole procedure is repeated for the new blocks A and B . 22 22 When SORT = 'S', the diagonal blocks of the generalized real Schur form are reordered before each step of the reduction, so that each cluster of generalized eigenvalues, defined as specified in the definition of TOL, appears in adjacent blocks. The blocks for each cluster are merged together, and the procedure described above is applied to the larger blocks. Using the option SORT = 'S' will usually provide better efficiency than the standard option (SORT = 'N'), proposed in [2], because there could be no or few unsuccessful attempts to compute individual transformation matrices X and Y of the form (1). However, the resulting dimensions of the blocks are usually larger; this could make subsequent calculations less efficient. When SORT = 'C' or 'B', the procedure is similar to that for SORT = 'N' or 'S', respectively, but the blocks of A and B 22 22 whose eigenvalue(s) is (are) the closest to those of (A , B ) 11 11 (not to their mean) are selected and moved to the leading position of A and B . This is called the "closest-neighbour" strategy. 22 22References
[1] Kagstrom, B. and Westin, L. Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation. IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989. [2] Bavely, C. and Stewart, G.W. An Algorithm for Computing Reducing Subspaces by Block Diagonalization. SIAM J. Numer. Anal., 16, pp. 359-367, 1979. [3] Demmel, J. The Condition Number of Equivalence Transformations that Block Diagonalize Matrix Pencils. SIAM J. Numer. Anal., 20, pp. 599-610, 1983.Numerical Aspects
3 4 The algorithm usually requires 0(N ) operations, but 0(N ) are possible in the worst case, when the matrix pencil cannot be diagonalized by well-conditioned transformations.Further Comments
The individual non-orthogonal transformation matrices used in the reduction of A and B to a block-diagonal form have condition numbers of the order PMAX. This does not guarantee that their product is well-conditioned enough. The routine can be easily modified to provide estimates for the condition numbers of the clusters of generalized eigenvalues.Example
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