tgex2 - tgex2: reorder generalized Schur form

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subroutinectgex2(logicalwantq,logicalwantz,integern,complex,dimension(lda,*)a,integerlda,complex,dimension(ldb,*)b,integerldb,complex,dimension(ldq,*)q,integerldq,complex,dimension(ldz,*)z,integerldz,integerj1,integerinfo)CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation. Purpose: CTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) in an upper triangular matrix pair (A, B) by an unitary equivalence transformation. (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. Optionally, the matrices Q and Z of generalized Schur vectors are updated. Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H ParametersWANTQ WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. WANTZ WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. N N is INTEGER The order of the matrices A and B. N >= 0. A A is COMPLEX array, dimension (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX array, dimension (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Q Q is COMPLEX array, dimension (LDQ,N) If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE.. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N. Z Z is COMPLEX array, dimension (LDZ,N) If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE.. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N. J1 J1 is INTEGER The index to the first block (A11, B11). INFO INFO is INTEGER =0: Successful exit. =1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill- conditioned. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. References: [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. subroutinedtgex2(logicalwantq,logicalwantz,integern,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecision,dimension(ldq,*)q,integerldq,doubleprecision,dimension(ldz,*)z,integerldz,integerj1,integern1,integern2,doubleprecision,dimension(*)work,integerlwork,integerinfo)DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. Purpose: DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation. (A, B) must be in generalized real Schur canonical form (as returned by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular. Optionally, the matrices Q and Z of generalized Schur vectors are updated. Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T ParametersWANTQ WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. WANTZ WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. N N is INTEGER The order of the matrices A and B. N >= 0. A A is DOUBLE PRECISION array, dimensions (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is DOUBLE PRECISION array, dimensions (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Q Q is DOUBLE PRECISION array, dimension (LDQ,N) On entry, if WANTQ = .TRUE., the orthogonal matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE.. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N. Z Z is DOUBLE PRECISION array, dimension (LDZ,N) On entry, if WANTZ =.TRUE., the orthogonal matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE.. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N. J1 J1 is INTEGER The index to the first block (A11, B11). 1 <= J1 <= N. N1 N1 is INTEGER The order of the first block (A11, B11). N1 = 0, 1 or 2. N2 N2 is INTEGER The order of the second block (A22, B22). N2 = 0, 1 or 2. WORK WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 ) INFO INFO is INTEGER =0: Successful exit >0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too ill-conditioned. <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1). Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. References: [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. subroutinestgex2(logicalwantq,logicalwantz,integern,real,dimension(lda,*)a,integerlda,real,dimension(ldb,*)b,integerldb,real,dimension(ldq,*)q,integerldq,real,dimension(ldz,*)z,integerldz,integerj1,integern1,integern2,real,dimension(*)work,integerlwork,integerinfo)STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. Purpose: STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation. (A, B) must be in generalized real Schur canonical form (as returned by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular. Optionally, the matrices Q and Z of generalized Schur vectors are updated. Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T ParametersWANTQ WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. WANTZ WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. N N is INTEGER The order of the matrices A and B. N >= 0. A A is REAL array, dimension (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is REAL array, dimension (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Q Q is REAL array, dimension (LDQ,N) On entry, if WANTQ = .TRUE., the orthogonal matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE.. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N. Z Z is REAL array, dimension (LDZ,N) On entry, if WANTZ =.TRUE., the orthogonal matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE.. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N. J1 J1 is INTEGER The index to the first block (A11, B11). 1 <= J1 <= N. N1 N1 is INTEGER The order of the first block (A11, B11). N1 = 0, 1 or 2. N2 N2 is INTEGER The order of the second block (A22, B22). N2 = 0, 1 or 2. WORK WORK is REAL array, dimension (MAX(1,LWORK)). LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 ) INFO INFO is INTEGER =0: Successful exit >0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too ill-conditioned. <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1). Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. References: [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. subroutineztgex2(logicalwantq,logicalwantz,integern,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldb,*)b,integerldb,complex*16,dimension(ldq,*)q,integerldq,complex*16,dimension(ldz,*)z,integerldz,integerj1,integerinfo)ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation. Purpose: ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) in an upper triangular matrix pair (A, B) by an unitary equivalence transformation. (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. Optionally, the matrices Q and Z of generalized Schur vectors are updated. Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H ParametersWANTQ WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. WANTZ WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. N N is INTEGER The order of the matrices A and B. N >= 0. A A is COMPLEX*16 array, dimensions (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimensions (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Q Q is COMPLEX*16 array, dimension (LDQ,N) If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE.. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N. Z Z is COMPLEX*16 array, dimension (LDZ,N) If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE.. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N. J1 J1 is INTEGER The index to the first block (A11, B11). INFO INFO is INTEGER =0: Successful exit. =1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill- conditioned. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. References: [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

tgex2 - tgex2: reorder generalized Schur form

Functions subroutine ctgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info) CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation. subroutine dtgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info) DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. subroutine stgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info) STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. subroutine ztgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info) ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.