**Purpose**

To reduce a matrix A in real Schur form to a block-diagonal form using well-conditioned non-orthogonal similarity transformations. The condition numbers of the transformations used for reduction are roughly bounded by PMAX*PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in a given matrix X. The real Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same block.

SUBROUTINE MB03RD( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS, $ BLSIZE, WR, WI, TOL, DWORK, INFO ) C .. Scalar Arguments .. CHARACTER JOBX, SORT INTEGER INFO, LDA, LDX, N, NBLCKS DOUBLE PRECISION PMAX, TOL C .. Array Arguments .. INTEGER BLSIZE(*) DOUBLE PRECISION A(LDA,*), DWORK(*), WI(*), WR(*), X(LDX,*)

**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 blocks of the 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 block; = '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).

N (input) INTEGER The order of the matrices A and X. N >= 0. PMAX (input) DOUBLE PRECISION An upper bound for the infinity norm of elementary submatrices of the individual transformations used for reduction (see METHOD). PMAX >= 1.0D0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the matrix A to be block-diagonalized, in real Schur form. On exit, the leading N-by-N part of this array contains the computed block-diagonal matrix, in real Schur canonical form. The non-diagonal blocks are set to zero. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). X (input/output) DOUBLE PRECISION array, dimension (LDX,N) 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-orthogonal 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. WR, (output) DOUBLE PRECISION arrays, dimension (N) WI These arrays contain the real and imaginary parts, respectively, of the eigenvalues of the matrix A.

TOL DOUBLE PRECISION The tolerance to be used in the ordering of the diagonal blocks of the real Schur form matrix. If the user sets TOL > 0, then the given value of TOL is used as an absolute tolerance: a block i and a temporarily fixed block 1 (the first block of the current trailing submatrix to be reduced) are considered to belong to the same cluster if their eigenvalues satisfy | lambda_1 - lambda_i | <= TOL. If the user sets TOL < 0, then the given value of TOL is used as a relative tolerance: a block i and a temporarily fixed block 1 are considered to belong to the same cluster if their eigenvalues 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.

DWORK DOUBLE PRECISION array, dimension (N)

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Consider first that SORT = 'N'. Let ( A A ) ( 11 12 ) A = ( ), ( 0 A ) ( 22 ) be the given matrix in real Schur form, where initially A is the 11 first diagonal block of dimension 1-by-1 or 2-by-2. 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, an 1-by-1 (or 2-by-2) diagonal block of A , whose eigenvalue(s) is (are) the closest to the mean of those 22 of A is selected, and moved by orthogonal similarity 11 transformations in the leading position of A ; the moved diagonal 22 block is then added to the block A , increasing its order by 1 11 (or 2). 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 matrix A . 22 When SORT = 'S', the diagonal blocks of the real 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 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 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.

[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.

3 4 The algorithm usually requires 0(N ) operations, but 0(N ) are possible in the worst case, when all diagonal blocks in the real Schur form of A are 1-by-1, and the matrix cannot be diagonalized by well-conditioned transformations.

The individual non-orthogonal transformation matrices used in the reduction of A to a block-diagonal form have condition numbers of the order PMAX*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.

**Program Text**

* MB03RD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX PARAMETER ( NMAX = 10 ) INTEGER LDA, LDX PARAMETER ( LDA = NMAX, LDX = NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = 3*NMAX ) * .. Local Scalars .. CHARACTER*1 JOBX, SORT INTEGER I, INFO, J, N, NBLCKS, SDIM DOUBLE PRECISION PMAX, TOL * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), WI(NMAX), WR(NMAX), $ X(LDX,NMAX) INTEGER BLSIZE(NMAX) LOGICAL BWORK(NMAX) * .. External Functions .. LOGICAL SELECT * .. External Subroutines .. EXTERNAL DGEES, MB03RD * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) N, PMAX, TOL, JOBX, SORT IF ( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99972 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N ) * Compute Schur form, eigenvalues and Schur vectors. CALL DGEES( 'Vectors', 'Not sorted', SELECT, N, A, LDA, SDIM, $ WR, WI, X, LDX, DWORK, LDWORK, BWORK, INFO ) IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE * Block-diagonalization. CALL MB03RD( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS, $ BLSIZE, WR, WI, TOL, DWORK, INFO ) IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99997 ) INFO ELSE WRITE ( NOUT, FMT = 99995 ) NBLCKS WRITE ( NOUT, FMT = 99994 ) ( BLSIZE(I), I = 1,NBLCKS ) WRITE ( NOUT, FMT = 99993 ) DO 10 I = 1, N WRITE ( NOUT, FMT = 99992 ) ( A(I,J), J = 1,N ) 10 CONTINUE WRITE ( NOUT, FMT = 99991 ) DO 20 I = 1, N WRITE ( NOUT, FMT = 99992 ) ( X(I,J), J = 1,N ) 20 CONTINUE END IF END IF END IF * STOP * 99999 FORMAT (' MB03RD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from DGEES = ',I2) 99997 FORMAT (' INFO on exit from MB03RD = ',I2) 99995 FORMAT (' The number of blocks is ',I5) 99994 FORMAT (' The orders of blocks are ',/(20(I3,2X))) 99993 FORMAT (' The block-diagonal matrix is ') 99992 FORMAT (8X,20(1X,F8.4)) 99991 FORMAT (' The transformation matrix is ') 99972 FORMAT (/' N is out of range.',/' N = ',I5) END

MB03RD EXAMPLE PROGRAM DATA 8 1.D03 1.D-2 U S 1. -1. 1. 2. 3. 1. 2. 3. 1. 1. 3. 4. 2. 3. 4. 2. 0. 0. 1. -1. 1. 5. 4. 1. 0. 0. 0. 1. -1. 3. 1. 2. 0. 0. 0. 1. 1. 2. 3. -1. 0. 0. 0. 0. 0. 1. 5. 1. 0. 0. 0. 0. 0. 0. 0.99999999 -0.99999999 0. 0. 0. 0. 0. 0. 0.99999999 0.99999999

MB03RD EXAMPLE PROGRAM RESULTS The number of blocks is 2 The orders of blocks are 6 2 The block-diagonal matrix is 1.0000 -1.0000 -1.2247 -0.7071 -3.4186 1.4577 0.0000 0.0000 1.0000 1.0000 0.0000 1.4142 -5.1390 3.1637 0.0000 0.0000 0.0000 0.0000 1.0000 -1.7321 -0.0016 2.0701 0.0000 0.0000 0.0000 0.0000 0.5774 1.0000 0.7516 1.1379 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 -5.8606 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1706 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 -0.8850 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 The transformation matrix is 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9045 0.1957 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 -0.3015 0.9755 0.0000 0.0000 0.8165 0.0000 -0.5768 -0.0156 -0.3015 0.0148 0.0000 0.0000 -0.4082 0.7071 -0.5768 -0.0156 0.0000 -0.0534 0.0000 0.0000 -0.4082 -0.7071 -0.5768 -0.0156 0.0000 0.0801 0.0000 0.0000 0.0000 0.0000 -0.0276 0.9805 0.0000 0.0267 0.0000 0.0000 0.0000 0.0000 0.0332 -0.0066 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0011 0.1948 0.0000 0.0000