CTGEXC reorders the generalized Schur decomposition of a complex


 matrix pair (A,B), using an unitary equivalence transformation


 (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with


 row index IFST is moved to row ILST.


 (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

WANTQ

          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 upper triangular 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 upper triangular 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)


          On entry, if WANTQ = .TRUE., the unitary matrix Q.


          On exit, the updated matrix Q.


          If WANTQ = .FALSE., Q is not referenced.

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)


          On entry, if WANTZ = .TRUE., the unitary matrix Z.


          On exit, the updated matrix Z.


          If WANTZ = .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1;


          If WANTZ = .TRUE., LDZ >= N.

IFST

          IFST is INTEGER

ILST

          ILST is INTEGER


          Specify the reordering of the diagonal blocks of (A, B).


          The block with row index IFST is moved to row ILST, by a


          sequence of swapping between adjacent blocks.

INFO

          INFO is INTEGER


           =0:  Successful exit.


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


           =1:  The transformed matrix pair (A, B) would be too far


                from generalized Schur form; the problem is ill-


                conditioned. (A, B) may have been partially reordered,


                and ILST points to the first row of the current


                position of the block being moved.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

[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.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

 DTGEXC reorders the generalized real Schur decomposition of a real


 matrix pair (A,B) using an orthogonal equivalence transformation


                (A, B) = Q * (A, B) * Z**T,


 so that the diagonal block of (A, B) with row index IFST is moved


 to row ILST.


 (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

WANTQ

          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, dimension (LDA,N)


          On entry, the matrix A in generalized real Schur canonical


          form.


          On exit, the updated matrix A, again in generalized


          real Schur canonical form.

LDA

          LDA is INTEGER


          The leading dimension of the array A. LDA >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (LDB,N)


          On entry, the matrix B in generalized real Schur canonical


          form (A,B).


          On exit, the updated matrix B, again in generalized


          real Schur canonical form (A,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.


          If WANTQ = .FALSE., Q is not referenced.

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.


          If WANTZ = .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1.


          If WANTZ = .TRUE., LDZ >= N.

IFST

          IFST is INTEGER

ILST

          ILST is INTEGER


          Specify the reordering of the diagonal blocks of (A, B).


          The block with row index IFST is moved to row ILST, by a


          sequence of swapping between adjacent blocks.


          On exit, if IFST pointed on entry to the second row of


          a 2-by-2 block, it is changed to point to the first row;


          ILST always points to the first row of the block in its


          final position (which may differ from its input value by


          +1 or -1). 1 <= IFST, ILST <= N.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


           =0:  successful exit.


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


           =1:  The transformed matrix pair (A, B) would be too far


                from generalized Schur form; the problem is ill-


                conditioned. (A, B) may have been partially reordered,


                and ILST points to the first row of the current


                position of the block being moved.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

  [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.

 STGEXC reorders the generalized real Schur decomposition of a real


 matrix pair (A,B) using an orthogonal equivalence transformation


                (A, B) = Q * (A, B) * Z**T,


 so that the diagonal block of (A, B) with row index IFST is moved


 to row ILST.


 (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

WANTQ

          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 generalized real Schur canonical


          form.


          On exit, the updated matrix A, again in generalized


          real Schur canonical form.

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 generalized real Schur canonical


          form (A,B).


          On exit, the updated matrix B, again in generalized


          real Schur canonical form (A,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.


          If WANTQ = .FALSE., Q is not referenced.

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.


          If WANTZ = .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1.


          If WANTZ = .TRUE., LDZ >= N.

IFST

          IFST is INTEGER

ILST

          ILST is INTEGER


          Specify the reordering of the diagonal blocks of (A, B).


          The block with row index IFST is moved to row ILST, by a


          sequence of swapping between adjacent blocks.


          On exit, if IFST pointed on entry to the second row of


          a 2-by-2 block, it is changed to point to the first row;


          ILST always points to the first row of the block in its


          final position (which may differ from its input value by


          +1 or -1). 1 <= IFST, ILST <= N.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


           =0:  successful exit.


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


           =1:  The transformed matrix pair (A, B) would be too far


                from generalized Schur form; the problem is ill-


                conditioned. (A, B) may have been partially reordered,


                and ILST points to the first row of the current


                position of the block being moved.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

  [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.

 ZTGEXC reorders the generalized Schur decomposition of a complex


 matrix pair (A,B), using an unitary equivalence transformation


 (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with


 row index IFST is moved to row ILST.


 (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

WANTQ

          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, dimension (LDA,N)


          On entry, the upper triangular 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, dimension (LDB,N)


          On entry, the upper triangular 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)


          On entry, if WANTQ = .TRUE., the unitary matrix Q.


          On exit, the updated matrix Q.


          If WANTQ = .FALSE., Q is not referenced.

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)


          On entry, if WANTZ = .TRUE., the unitary matrix Z.


          On exit, the updated matrix Z.


          If WANTZ = .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1;


          If WANTZ = .TRUE., LDZ >= N.

IFST

          IFST is INTEGER

ILST

          ILST is INTEGER


          Specify the reordering of the diagonal blocks of (A, B).


          The block with row index IFST is moved to row ILST, by a


          sequence of swapping between adjacent blocks.

INFO

          INFO is INTEGER


           =0:  Successful exit.


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


           =1:  The transformed matrix pair (A, B) would be too far


                from generalized Schur form; the problem is ill-


                conditioned. (A, B) may have been partially reordered,


                and ILST points to the first row of the current


                position of the block being moved.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

[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.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.