Purpose
To reduce a complex matrix pair (A,B) in generalized complex Schur form to a block-diagonal form using well-conditioned non-unitary 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 MB04RZ( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X, $ LDX, Y, LDY, NBLCKS, BLSIZE, ALPHA, BETA, $ TOL, IWORK, INFO ) C .. Scalar Arguments .. CHARACTER JOBX, JOBY, SORT INTEGER INFO, LDA, LDB, LDX, LDY, N, NBLCKS DOUBLE PRECISION PMAX, TOL C .. Array Arguments .. INTEGER BLSIZE(*), IWORK(*) COMPLEX*16 A(LDA,*), ALPHA(*), B(LDB,*), BETA(*), 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 elements of the generalized Schur form are reordered, as follows: = 'N': The diagonal elements are not reordered; = 'S': The diagonal elements are reordered before each step of reduction, so that clustered eigenvalues appear in the same pair of blocks. = 'C': The diagonal elements are not reordered, but the "closest-neighbour" strategy is used instead of the standard "closest to the mean" strategy (see METHOD); = 'B': The diagonal elements 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) COMPLEX*16 array, dimension (LDA,N) On entry, the leading N-by-N upper triangular part of this array must contain the upper triangular matrix A in the generalized complex Schur form, as returned by the LAPACK Library routine ZGGES. The strictly lower triangular part is used as workspace. On exit, the leading N-by-N upper triangular part of this array contains the computed block-diagonal matrix, 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) COMPLEX*16 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 complex Schur form, as returned by the LAPACK Library routine ZGGES. The diagonal elements of B are real non-negative, hence the imaginary parts of these elements are zero. 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 real non-negative. LDB INTEGER The leading dimension of the array B. LDB >= MAX(1,N). X (input/output) COMPLEX*16 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 ZGGES. 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-unitary 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) COMPLEX*16 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 ZGGES. 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-unitary 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. ALPHA (output) COMPLEX*16 array, dimension (N) BETA (output) COMPLEX*16 array, dimension (N) If INFO = 0, then ALPHA(j)/BETA(j), j = 1, ..., N, are the generalized eigenvalues of the matrix pair (A, B) (the diagonals of the complex Schur form). All BETA(j) are non-negative real numbers. The quotients ALPHA(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 ZGGES, 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 elements of the upper triangular matrix pair. If the user sets TOL > 0, then the given value of TOL is used as an absolute tolerance: an eigenvalue i and a temporarily fixed eigenvalue 1 (represented by the first element of the current trailing pair of submatrices to be reduced) are considered to belong to the same cluster if they 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: an eigenvalue i and a temporarily fixed eigenvalue 1 are considered to belong to the same cluster if they 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+2)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 Schur form, where initially A and B are the first diagonal elements. 11 11 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 conjugate-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, an eigenvalue of (A , B ) closest to the mean of those of (A , B ) is selected, 22 22 11 11 and moved by unitary equivalence transformations in the leading position of (A , B ); the moved diagonal elements in A and B are 22 22 then added to the blocks A and B , respectively, increasing 11 11 their order by 1. 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 transformation 11 22 11 22 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 elements of the generalized 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 elements. The elements 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-unitary 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. For efficiency reasons, the "absolute value" (or "magnitude") of the complex elements x of the transformation matrices is computed as |real(x)| + |imag(x)|.Example
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