Purpose
To reduce an upper triangular complex matrix A (Schur form) to a block-diagonal form using well-conditioned non-unitary similarity transformations. The condition numbers of the transformations used for reduction are roughly bounded by PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in a given matrix X. The Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same block.Specification
SUBROUTINE MB03RZ( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS, $ BLSIZE, W, TOL, INFO )C .. Scalar Arguments .. CHARACTER JOBX, SORT INTEGER INFO, LDA, LDX, N, NBLCKS DOUBLE PRECISION PMAX, TOL C .. Array Arguments .. INTEGER BLSIZE(*) COMPLEX*16 A(LDA,*), W(*), X(LDX,*)Arguments
Mode Parameters
JOBX CHARACTER*1 Specifies whether or not the transformations are accumulated, as follows: = 'N': The transformations are not accumulated; = 'U': The transformations are accumulated in X (the given matrix X is updated). SORT CHARACTER*1 Specifies whether or not the diagonal elements of the 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 block; = '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 and X. 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). 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 to be block-diagonalized. On exit, the leading N-by-N upper triangular part of this array contains the computed block-diagonal matrix, in Schur form. The strictly lower triangular part is used as workspace, but it is set to zero before exit. LDA INTEGER The leading dimension of array A. LDA >= 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. On exit, if JOBX = 'U', the leading N-by-N part of this array contains the product of the given matrix X and the transformation matrix that reduced A to block-diagonal form. The transformation matrix is itself a product of non-unitary similarity 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 array X. LDX >= 1, if JOBX = 'N'; LDX >= MAX(1,N), if JOBX = 'U'. NBLCKS (output) INTEGER The number of diagonal blocks of the matrix A. BLSIZE (output) INTEGER array, dimension (N) The first NBLCKS elements of this array contain the orders of the resulting diagonal blocks of the matrix A. W (output) COMPLEX*16 array, dimension (N) This array contains the eigenvalues of the matrix A.Tolerances
TOL DOUBLE PRECISION The tolerance to be used in the ordering of the diagonal elements of the upper triangular matrix. 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 (the first element of the current trailing submatrix to be reduced) are considered to belong to the same cluster if they satisfy | 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 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). If SORT = 'N' or 'C', this parameter is not referenced.Error Indicator
INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.Method
Consider first that SORT = 'N'. Let ( A A ) ( 11 12 ) A = ( ), ( 0 A ) ( 22 ) be the given matrix in Schur form, where initially A is the 11 first diagonal element. An attempt is made to compute a transformation matrix X of the form ( I P ) X = ( ) (1) ( 0 I ) (partitioned as A), so that ( A 0 ) -1 ( 11 ) X A X = ( ), ( 0 A ) ( 22 ) and the elements of P do not exceed the value PMAX in magnitude. An adaptation of the standard method for solving Sylvester equations [1], which controls the magnitude of the individual elements of the computed solution [2], is used to obtain matrix P. When this attempt failed, a diagonal element of A , closest to 22 the mean of those of A is selected, and moved by unitary 11 similarity transformations in the leading position of A ; the 22 moved diagonal element is then added to the block A , increasing 11 its order by 1. Another attempt is made to compute a suitable transformation matrix X with the new definitions of the blocks A 11 and A . After a successful transformation matrix X has been 22 obtained, it postmultiplies the current transformation matrix (if JOBX = 'U'), and the whole procedure is repeated for the block A . 22 When SORT = 'S', the diagonal elements of the Schur form are reordered before each step of the reduction, so that each cluster of 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 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 block of A whose 22 eigenvalue(s) is (are) the closest to those of A (not to their 11 mean) is selected and moved to the leading position of A . This 22 is called the "closest-neighbour" strategy.References
[1] Bartels, R.H. and Stewart, G.W. T Solution of the matrix equation A X + XB = C. Comm. A.C.M., 15, pp. 820-826, 1972. [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 cannot be diagonalized by well-conditioned transformations.Further Comments
The individual non-unitary transformation matrices used in the reduction of A 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 eigenvalues.Example
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