**Purpose**

To compute orthogonal matrices Q1, Q2, Q3 for a real 2-by-2, 3-by-3, or 4-by-4 regular block upper triangular pencil ( A11 A12 ) ( B11 B12 ) ( D11 D12 ) aAB - bD = a ( ) ( ) - b ( ), (1) ( 0 A22 ) ( 0 B22 ) ( 0 D22 ) such that the pencil a(Q3' A Q2 )(Q2' B Q1 ) - b(Q3' D Q1) is still in block upper triangular form, but the eigenvalues in Spec(A11 B11, D11), Spec(A22 B22, D22) are exchanged, where Spec(X,Y) denotes the spectrum of the matrix pencil (X,Y), and M' denotes the transpose of the matrix M. Optionally, to upper triangularize the real regular pencil in block lower triangular form ( A11 0 ) ( B11 0 ) ( D11 0 ) aAB - bD = a ( ) ( ) - b ( ), (2) ( A21 A22 ) ( B21 B22 ) ( D21 D22 ) while keeping the eigenvalues in the same diagonal position.

SUBROUTINE MB03CD( UPLO, N1, N2, PREC, A, LDA, B, LDB, D, LDD, Q1, $ LDQ1, Q2, LDQ2, Q3, LDQ3, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK, $ N1, N2 DOUBLE PRECISION PREC C .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( LDD, * ), $ DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ), $ Q3( LDQ3, * )

**Mode Parameters**

UPLO CHARACTER*1 Specifies if the pencil is in lower or upper block triangular form on entry, as follows: = 'U': Upper block triangular, eigenvalues are exchanged on exit; = 'L': Lower block triangular, eigenvalues are not exchanged on exit.

N1 (input/output) INTEGER Size of the upper left block, N1 <= 2. If UPLO = 'U' and INFO = 0, or UPLO = 'L' and INFO <> 0, N1 and N2 are exchanged on exit; otherwise, N1 is unchanged on exit. N2 (input/output) INTEGER Size of the lower right block, N2 <= 2. If UPLO = 'U' and INFO = 0, or UPLO = 'L' and INFO <> 0, N1 and N2 are exchanged on exit; otherwise, N2 is unchanged on exit. PREC (input) DOUBLE PRECISION The machine precision, (relative machine precision)*base. See the LAPACK Library routine DLAMCH. A (input or input/output) DOUBLE PRECISION array, dimension (LDA, N1+N2) On entry, the leading (N1+N2)-by-(N1+N2) part of this array must contain the matrix A of the pencil aAB - bD. The (2,1) block, if UPLO = 'U', or the (1,2) block, if UPLO = 'L', need not be set to zero. On exit, if N1 = N2 = 1, this array contains the matrix [ 0 1 ] J' A J, where J = [ -1 0 ]; otherwise, this array is unchanged on exit. LDA INTEGER The leading dimension of the array A. LDA >= N1+N2. B (input or input/output) DOUBLE PRECISION array, dimension (LDB, N1+N2) On entry, the leading (N1+N2)-by-(N1+N2) part of this array must contain the matrix B of the pencil aAB - bD. The (2,1) block, if UPLO = 'U', or the (1,2) block, if UPLO = 'L', need not be set to zero. On exit, if N1 = N2 = 1, this array contains the matrix J' B J; otherwise, this array is unchanged on exit. LDB INTEGER The leading dimension of the array B. LDB >= N1+N2. D (input/output) DOUBLE PRECISION array, dimension (LDD, N1+N2) On entry, the leading (N1+N2)-by-(N1+N2) part of this array must contain the matrix D of the pencil aAB - bD. On exit, if N1 = 2 or N2 = 2, the leading (N1+N2)-by-(N1+N2) part of this array contains the transformed matrix D in real Schur form. If N1 = 1 and N2 = 1, this array contains the matrix J' D J. LDD INTEGER The leading dimension of the array D. LDD >= N1+N2. Q1 (output) DOUBLE PRECISION array, dimension (LDQ1, N1+N2) The leading (N1+N2)-by-(N1+N2) part of this array contains the first orthogonal transformation matrix. LDQ1 INTEGER The leading dimension of the array Q1. LDQ1 >= N1+N2. Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, N1+N2) The leading (N1+N2)-by-(N1+N2) part of this array contains the second orthogonal transformation matrix. LDQ2 INTEGER The leading dimension of the array Q2. LDQ2 >= N1+N2. Q3 (output) DOUBLE PRECISION array, dimension (LDQ3, N1+N2) The leading (N1+N2)-by-(N1+N2) part of this array contains the third orthogonal transformation matrix. LDQ3 INTEGER The leading dimension of the array Q3. LDQ3 >= N1+N2.

DWORK DOUBLE PRECISION array, dimension (LDWORK) If N1+N2 = 2 then DWORK is not referenced. LDWORK INTEGER The dimension of the array DWORK. If N1+N2 = 2, then LDWORK = 0; otherwise, LDWORK >= 16*N1 + 10*N2 + 23, UPLO = 'U'; LDWORK >= 10*N1 + 16*N2 + 23, UPLO = 'L'.

INFO INTEGER = 0: succesful exit; = 1: the QZ iteration failed in the LAPACK routine DGGEV; = 2: another error occured while executing a routine in DGGEV; = 3: the QZ iteration failed in the LAPACK routine DGGES; = 4: another error occured during execution of DGGES; = 5: reordering of aA*B - bD in the LAPACK routine DTGSEN failed because the transformed matrix pencil aA*B - bD would be too far from generalized Schur form; the problem is very ill-conditioned.

The algorithm uses orthogonal transformations as described in [2] (page 21). The QZ algorithm is used for N1 = 2 or N2 = 2, but it always acts on an upper block triangular pencil.

[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H. Numerical computation of deflating subspaces of skew- Hamiltonian/Hamiltonian pencils. SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002. [2] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H. Numerical Solution of Real Skew-Hamiltonian/Hamiltonian Eigenproblems. Tech. Rep., Technical University Chemnitz, Germany, Nov. 2007.

The algorithm is numerically backward stable.

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**Program Text**

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