DOUBLE PRECISION routines for triangular matrices, generalized problem (i.e., a pair of triangular matrices) matrix
dtgevc
USAGE:
m, info, vl, vr = NumRu::Lapack.dtgevc( side, howmny, select, s, p, vl, vr, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO )
* Purpose
* =======
*
* DTGEVC computes some or all of the right and/or left eigenvectors of
* a pair of real matrices (S,P), where S is a quasi-triangular matrix
* and P is upper triangular. Matrix pairs of this type are produced by
* the generalized Schur factorization of a matrix pair (A,B):
*
* A = Q*S*Z**T, B = Q*P*Z**T
*
* as computed by DGGHRD + DHGEQZ.
*
* The right eigenvector x and the left eigenvector y of (S,P)
* corresponding to an eigenvalue w are defined by:
*
* S*x = w*P*x, (y**H)*S = w*(y**H)*P,
*
* where y**H denotes the conjugate tranpose of y.
* The eigenvalues are not input to this routine, but are computed
* directly from the diagonal blocks of S and P.
*
* This routine returns the matrices X and/or Y of right and left
* eigenvectors of (S,P), or the products Z*X and/or Q*Y,
* where Z and Q are input matrices.
* If Q and Z are the orthogonal factors from the generalized Schur
* factorization of a matrix pair (A,B), then Z*X and Q*Y
* are the matrices of right and left eigenvectors of (A,B).
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* = 'R': compute right eigenvectors only;
* = 'L': compute left eigenvectors only;
* = 'B': compute both right and left eigenvectors.
*
* HOWMNY (input) CHARACTER*1
* = 'A': compute all right and/or left eigenvectors;
* = 'B': compute all right and/or left eigenvectors,
* backtransformed by the matrices in VR and/or VL;
* = 'S': compute selected right and/or left eigenvectors,
* specified by the logical array SELECT.
*
* SELECT (input) LOGICAL array, dimension (N)
* If HOWMNY='S', SELECT specifies the eigenvectors to be
* computed. If w(j) is a real eigenvalue, the corresponding
* real eigenvector is computed if SELECT(j) is .TRUE..
* If w(j) and w(j+1) are the real and imaginary parts of a
* complex eigenvalue, the corresponding complex eigenvector
* is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
* and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
* set to .FALSE..
* Not referenced if HOWMNY = 'A' or 'B'.
*
* N (input) INTEGER
* The order of the matrices S and P. N >= 0.
*
* S (input) DOUBLE PRECISION array, dimension (LDS,N)
* The upper quasi-triangular matrix S from a generalized Schur
* factorization, as computed by DHGEQZ.
*
* LDS (input) INTEGER
* The leading dimension of array S. LDS >= max(1,N).
*
* P (input) DOUBLE PRECISION array, dimension (LDP,N)
* The upper triangular matrix P from a generalized Schur
* factorization, as computed by DHGEQZ.
* 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
* of S must be in positive diagonal form.
*
* LDP (input) INTEGER
* The leading dimension of array P. LDP >= max(1,N).
*
* VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
* On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
* contain an N-by-N matrix Q (usually the orthogonal matrix Q
* of left Schur vectors returned by DHGEQZ).
* On exit, if SIDE = 'L' or 'B', VL contains:
* if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
* if HOWMNY = 'B', the matrix Q*Y;
* if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
* SELECT, stored consecutively in the columns of
* VL, in the same order as their eigenvalues.
*
* A complex eigenvector corresponding to a complex eigenvalue
* is stored in two consecutive columns, the first holding the
* real part, and the second the imaginary part.
*
* Not referenced if SIDE = 'R'.
*
* LDVL (input) INTEGER
* The leading dimension of array VL. LDVL >= 1, and if
* SIDE = 'L' or 'B', LDVL >= N.
*
* VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
* On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
* contain an N-by-N matrix Z (usually the orthogonal matrix Z
* of right Schur vectors returned by DHGEQZ).
*
* On exit, if SIDE = 'R' or 'B', VR contains:
* if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
* if HOWMNY = 'B' or 'b', the matrix Z*X;
* if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
* specified by SELECT, stored consecutively in the
* columns of VR, in the same order as their
* eigenvalues.
*
* A complex eigenvector corresponding to a complex eigenvalue
* is stored in two consecutive columns, the first holding the
* real part and the second the imaginary part.
*
* Not referenced if SIDE = 'L'.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1, and if
* SIDE = 'R' or 'B', LDVR >= N.
*
* MM (input) INTEGER
* The number of columns in the arrays VL and/or VR. MM >= M.
*
* M (output) INTEGER
* The number of columns in the arrays VL and/or VR actually
* used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
* is set to N. Each selected real eigenvector occupies one
* column and each selected complex eigenvector occupies two
* columns.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
* eigenvalue.
*
* Further Details
* ===============
*
* Allocation of workspace:
* ---------- -- ---------
*
* WORK( j ) = 1-norm of j-th column of A, above the diagonal
* WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
* WORK( 2*N+1:3*N ) = real part of eigenvector
* WORK( 3*N+1:4*N ) = imaginary part of eigenvector
* WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
* WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
*
* Rowwise vs. columnwise solution methods:
* ------- -- ---------- -------- -------
*
* Finding a generalized eigenvector consists basically of solving the
* singular triangular system
*
* (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
*
* Consider finding the i-th right eigenvector (assume all eigenvalues
* are real). The equation to be solved is:
* n i
* 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
* k=j k=j
*
* where C = (A - w B) (The components v(i+1:n) are 0.)
*
* The "rowwise" method is:
*
* (1) v(i) := 1
* for j = i-1,. . .,1:
* i
* (2) compute s = - sum C(j,k) v(k) and
* k=j+1
*
* (3) v(j) := s / C(j,j)
*
* Step 2 is sometimes called the "dot product" step, since it is an
* inner product between the j-th row and the portion of the eigenvector
* that has been computed so far.
*
* The "columnwise" method consists basically in doing the sums
* for all the rows in parallel. As each v(j) is computed, the
* contribution of v(j) times the j-th column of C is added to the
* partial sums. Since FORTRAN arrays are stored columnwise, this has
* the advantage that at each step, the elements of C that are accessed
* are adjacent to one another, whereas with the rowwise method, the
* elements accessed at a step are spaced LDS (and LDP) words apart.
*
* When finding left eigenvectors, the matrix in question is the
* transpose of the one in storage, so the rowwise method then
* actually accesses columns of A and B at each step, and so is the
* preferred method.
*
* =====================================================================
*
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dtgex2
USAGE:
info, a, b, q, z = NumRu::Lapack.dtgex2( wantq, wantz, a, b, q, z, j1, n1, n2, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO )
* 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)' = Q(out) * A(out) * Z(out)'
* Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
*
*
* Arguments
* =========
*
* WANTQ (input) LOGICAL
* .TRUE. : update the left transformation matrix Q;
* .FALSE.: do not update Q.
*
* WANTZ (input) LOGICAL
* .TRUE. : update the right transformation matrix Z;
* .FALSE.: do not update Z.
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimensions (LDA,N)
* On entry, the matrix A in the pair (A, B).
* On exit, the updated matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) DOUBLE PRECISION array, dimensions (LDB,N)
* On entry, the matrix B in the pair (A, B).
* On exit, the updated matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* Q (input/output) 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 (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1.
* If WANTQ = .TRUE., LDQ >= N.
*
* Z (input/output) 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 (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If WANTZ = .TRUE., LDZ >= N.
*
* J1 (input) INTEGER
* The index to the first block (A11, B11). 1 <= J1 <= N.
*
* N1 (input) INTEGER
* The order of the first block (A11, B11). N1 = 0, 1 or 2.
*
* N2 (input) INTEGER
* The order of the second block (A22, B22). N2 = 0, 1 or 2.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
*
* INFO (output) 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).
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* 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.
*
* [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.
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
* loops. Sven Hammarling, 1/5/02.
*
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dtgexc
USAGE:
work, info, a, b, q, z, ifst, ilst = NumRu::Lapack.dtgexc( wantq, wantz, a, b, q, z, ifst, ilst, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO )
* Purpose
* =======
*
* 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',
*
* 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)' = Q(out) * A(out) * Z(out)'
* Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
*
*
* Arguments
* =========
*
* WANTQ (input) LOGICAL
* .TRUE. : update the left transformation matrix Q;
* .FALSE.: do not update Q.
*
* WANTZ (input) LOGICAL
* .TRUE. : update the right transformation matrix Z;
* .FALSE.: do not update Z.
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* A (input/output) 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 (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) 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 (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* Q (input/output) 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 (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1.
* If WANTQ = .TRUE., LDQ >= N.
*
* Z (input/output) 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 (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If WANTZ = .TRUE., LDZ >= N.
*
* IFST (input/output) INTEGER
* ILST (input/output) 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 (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) 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 (output) 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.
*
* Further Details
* ===============
*
* Based on contributions by
* 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.
*
* =====================================================================
*
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dtgsen
USAGE:
alphar, alphai, beta, m, pl, pr, dif, work, iwork, info, a, b, q, z = NumRu::Lapack.dtgsen( ijob, wantq, wantz, select, a, b, q, z, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
* Purpose
* =======
*
* DTGSEN reorders the generalized real Schur decomposition of a real
* matrix pair (A, B) (in terms of an orthonormal equivalence trans-
* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
* appears in the leading diagonal blocks of the upper quasi-triangular
* matrix A and the upper triangular B. The leading columns of Q and
* Z form orthonormal bases of the corresponding left and right eigen-
* spaces (deflating subspaces). (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.
*
* DTGSEN also computes the generalized eigenvalues
*
* w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
*
* of the reordered matrix pair (A, B).
*
* Optionally, DTGSEN computes the estimates of reciprocal condition
* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
* between the matrix pairs (A11, B11) and (A22,B22) that correspond to
* the selected cluster and the eigenvalues outside the cluster, resp.,
* and norms of "projections" onto left and right eigenspaces w.r.t.
* the selected cluster in the (1,1)-block.
*
* Arguments
* =========
*
* IJOB (input) INTEGER
* Specifies whether condition numbers are required for the
* cluster of eigenvalues (PL and PR) or the deflating subspaces
* (Difu and Difl):
* =0: Only reorder w.r.t. SELECT. No extras.
* =1: Reciprocal of norms of "projections" onto left and right
* eigenspaces w.r.t. the selected cluster (PL and PR).
* =2: Upper bounds on Difu and Difl. F-norm-based estimate
* (DIF(1:2)).
* =3: Estimate of Difu and Difl. 1-norm-based estimate
* (DIF(1:2)).
* About 5 times as expensive as IJOB = 2.
* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
* version to get it all.
* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
*
* WANTQ (input) LOGICAL
* .TRUE. : update the left transformation matrix Q;
* .FALSE.: do not update Q.
*
* WANTZ (input) LOGICAL
* .TRUE. : update the right transformation matrix Z;
* .FALSE.: do not update Z.
*
* SELECT (input) LOGICAL array, dimension (N)
* SELECT specifies the eigenvalues in the selected cluster.
* To select a real eigenvalue w(j), SELECT(j) must be set to
* .TRUE.. To select a complex conjugate pair of eigenvalues
* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
* either SELECT(j) or SELECT(j+1) or both must be set to
* .TRUE.; a complex conjugate pair of eigenvalues must be
* either both included in the cluster or both excluded.
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension(LDA,N)
* On entry, the upper quasi-triangular matrix A, with (A, B) in
* generalized real Schur canonical form.
* On exit, A is overwritten by the reordered matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) DOUBLE PRECISION array, dimension(LDB,N)
* On entry, the upper triangular matrix B, with (A, B) in
* generalized real Schur canonical form.
* On exit, B is overwritten by the reordered matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* ALPHAR (output) DOUBLE PRECISION array, dimension (N)
* ALPHAI (output) DOUBLE PRECISION array, dimension (N)
* BETA (output) DOUBLE PRECISION array, dimension (N)
* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
* and BETA(j),j=1,...,N are the diagonals of the complex Schur
* form (S,T) that would result if the 2-by-2 diagonal blocks of
* the real generalized Schur form of (A,B) were further reduced
* to triangular form using complex unitary transformations.
* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
* positive, then the j-th and (j+1)-st eigenvalues are a
* complex conjugate pair, with ALPHAI(j+1) negative.
*
* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
* On exit, Q has been postmultiplied by the left orthogonal
* transformation matrix which reorder (A, B); The leading M
* columns of Q form orthonormal bases for the specified pair of
* left eigenspaces (deflating subspaces).
* If WANTQ = .FALSE., Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1;
* and if WANTQ = .TRUE., LDQ >= N.
*
* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
* On exit, Z has been postmultiplied by the left orthogonal
* transformation matrix which reorder (A, B); The leading M
* columns of Z form orthonormal bases for the specified pair of
* left eigenspaces (deflating subspaces).
* If WANTZ = .FALSE., Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1;
* If WANTZ = .TRUE., LDZ >= N.
*
* M (output) INTEGER
* The dimension of the specified pair of left and right eigen-
* spaces (deflating subspaces). 0 <= M <= N.
*
* PL (output) DOUBLE PRECISION
* PR (output) DOUBLE PRECISION
* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
* reciprocal of the norm of "projections" onto left and right
* eigenspaces with respect to the selected cluster.
* 0 < PL, PR <= 1.
* If M = 0 or M = N, PL = PR = 1.
* If IJOB = 0, 2 or 3, PL and PR are not referenced.
*
* DIF (output) DOUBLE PRECISION array, dimension (2).
* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
* estimates of Difu and Difl.
* If M = 0 or N, DIF(1:2) = F-norm([A, B]).
* If IJOB = 0 or 1, DIF is not referenced.
*
* WORK (workspace/output) DOUBLE PRECISION array,
* dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 4*N+16.
* If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
* If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
*
* 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.
*
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= 1.
* If IJOB = 1, 2 or 4, LIWORK >= N+6.
* If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
*
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal size of the IWORK array,
* returns this value as the first entry of the IWORK array, and
* no error message related to LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* =0: Successful exit.
* <0: If INFO = -i, the i-th argument had an illegal value.
* =1: Reordering of (A, B) failed because the transformed
* matrix pair (A, B) would be too far from generalized
* Schur form; the problem is very ill-conditioned.
* (A, B) may have been partially reordered.
* If requested, 0 is returned in DIF(*), PL and PR.
*
* Further Details
* ===============
*
* DTGSEN first collects the selected eigenvalues by computing
* orthogonal U and W that move them to the top left corner of (A, B).
* In other words, the selected eigenvalues are the eigenvalues of
* (A11, B11) in:
*
* U'*(A, B)*W = (A11 A12) (B11 B12) n1
* ( 0 A22),( 0 B22) n2
* n1 n2 n1 n2
*
* where N = n1+n2 and U' means the transpose of U. The first n1 columns
* of U and W span the specified pair of left and right eigenspaces
* (deflating subspaces) of (A, B).
*
* If (A, B) has been obtained from the generalized real Schur
* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
* reordered generalized real Schur form of (C, D) is given by
*
* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
*
* and the first n1 columns of Q*U and Z*W span the corresponding
* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
*
* Note that if the selected eigenvalue is sufficiently ill-conditioned,
* then its value may differ significantly from its value before
* reordering.
*
* The reciprocal condition numbers of the left and right eigenspaces
* spanned by the first n1 columns of U and W (or Q*U and Z*W) may
* be returned in DIF(1:2), corresponding to Difu and Difl, resp.
*
* The Difu and Difl are defined as:
*
* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
* and
* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
*
* where sigma-min(Zu) is the smallest singular value of the
* (2*n1*n2)-by-(2*n1*n2) matrix
*
* Zu = [ kron(In2, A11) -kron(A22', In1) ]
* [ kron(In2, B11) -kron(B22', In1) ].
*
* Here, Inx is the identity matrix of size nx and A22' is the
* transpose of A22. kron(X, Y) is the Kronecker product between
* the matrices X and Y.
*
* When DIF(2) is small, small changes in (A, B) can cause large changes
* in the deflating subspace. An approximate (asymptotic) bound on the
* maximum angular error in the computed deflating subspaces is
*
* EPS * norm((A, B)) / DIF(2),
*
* where EPS is the machine precision.
*
* The reciprocal norm of the projectors on the left and right
* eigenspaces associated with (A11, B11) may be returned in PL and PR.
* They are computed as follows. First we compute L and R so that
* P*(A, B)*Q is block diagonal, where
*
* P = ( I -L ) n1 Q = ( I R ) n1
* ( 0 I ) n2 and ( 0 I ) n2
* n1 n2 n1 n2
*
* and (L, R) is the solution to the generalized Sylvester equation
*
* A11*R - L*A22 = -A12
* B11*R - L*B22 = -B12
*
* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
* An approximate (asymptotic) bound on the average absolute error of
* the selected eigenvalues is
*
* EPS * norm((A, B)) / PL.
*
* There are also global error bounds which valid for perturbations up
* to a certain restriction: A lower bound (x) on the smallest
* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
* (i.e. (A + E, B + F), is
*
* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
*
* An approximate bound on x can be computed from DIF(1:2), PL and PR.
*
* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
* (L', R') and unperturbed (L, R) left and right deflating subspaces
* associated with the selected cluster in the (1,1)-blocks can be
* bounded as
*
* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
*
* See LAPACK User's Guide section 4.11 or the following references
* for more information.
*
* Note that if the default method for computing the Frobenius-norm-
* based estimate DIF is not wanted (see DLATDF), then the parameter
* IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
* (IJOB = 2 will be used)). See DTGSYL for more details.
*
* Based on contributions by
* 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.
*
* [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.
*
* =====================================================================
*
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dtgsja
USAGE:
alpha, beta, ncycle, info, a, b, u, v, q = NumRu::Lapack.dtgsja( jobu, jobv, jobq, k, l, a, b, tola, tolb, u, v, q, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO )
* Purpose
* =======
*
* DTGSJA computes the generalized singular value decomposition (GSVD)
* of two real upper triangular (or trapezoidal) matrices A and B.
*
* On entry, it is assumed that matrices A and B have the following
* forms, which may be obtained by the preprocessing subroutine DGGSVP
* from a general M-by-N matrix A and P-by-N matrix B:
*
* N-K-L K L
* A = K ( 0 A12 A13 ) if M-K-L >= 0;
* L ( 0 0 A23 )
* M-K-L ( 0 0 0 )
*
* N-K-L K L
* A = K ( 0 A12 A13 ) if M-K-L < 0;
* M-K ( 0 0 A23 )
*
* N-K-L K L
* B = L ( 0 0 B13 )
* P-L ( 0 0 0 )
*
* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
* otherwise A23 is (M-K)-by-L upper trapezoidal.
*
* On exit,
*
* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
*
* where U, V and Q are orthogonal matrices, Z' denotes the transpose
* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
* ``diagonal'' matrices, which are of the following structures:
*
* If M-K-L >= 0,
*
* K L
* D1 = K ( I 0 )
* L ( 0 C )
* M-K-L ( 0 0 )
*
* K L
* D2 = L ( 0 S )
* P-L ( 0 0 )
*
* N-K-L K L
* ( 0 R ) = K ( 0 R11 R12 ) K
* L ( 0 0 R22 ) L
*
* where
*
* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
* S = diag( BETA(K+1), ... , BETA(K+L) ),
* C**2 + S**2 = I.
*
* R is stored in A(1:K+L,N-K-L+1:N) on exit.
*
* If M-K-L < 0,
*
* K M-K K+L-M
* D1 = K ( I 0 0 )
* M-K ( 0 C 0 )
*
* K M-K K+L-M
* D2 = M-K ( 0 S 0 )
* K+L-M ( 0 0 I )
* P-L ( 0 0 0 )
*
* N-K-L K M-K K+L-M
* ( 0 R ) = K ( 0 R11 R12 R13 )
* M-K ( 0 0 R22 R23 )
* K+L-M ( 0 0 0 R33 )
*
* where
* C = diag( ALPHA(K+1), ... , ALPHA(M) ),
* S = diag( BETA(K+1), ... , BETA(M) ),
* C**2 + S**2 = I.
*
* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
* ( 0 R22 R23 )
* in B(M-K+1:L,N+M-K-L+1:N) on exit.
*
* The computation of the orthogonal transformation matrices U, V or Q
* is optional. These matrices may either be formed explicitly, or they
* may be postmultiplied into input matrices U1, V1, or Q1.
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* = 'U': U must contain an orthogonal matrix U1 on entry, and
* the product U1*U is returned;
* = 'I': U is initialized to the unit matrix, and the
* orthogonal matrix U is returned;
* = 'N': U is not computed.
*
* JOBV (input) CHARACTER*1
* = 'V': V must contain an orthogonal matrix V1 on entry, and
* the product V1*V is returned;
* = 'I': V is initialized to the unit matrix, and the
* orthogonal matrix V is returned;
* = 'N': V is not computed.
*
* JOBQ (input) CHARACTER*1
* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
* the product Q1*Q is returned;
* = 'I': Q is initialized to the unit matrix, and the
* orthogonal matrix Q is returned;
* = 'N': Q is not computed.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* K (input) INTEGER
* L (input) INTEGER
* K and L specify the subblocks in the input matrices A and B:
* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
* of A and B, whose GSVD is going to be computed by DTGSJA.
* See Further Details.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
* matrix R or part of R. See Purpose for details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
* On entry, the P-by-N matrix B.
* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
* a part of R. See Purpose for details.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* TOLA (input) DOUBLE PRECISION
* TOLB (input) DOUBLE PRECISION
* TOLA and TOLB are the convergence criteria for the Jacobi-
* Kogbetliantz iteration procedure. Generally, they are the
* same as used in the preprocessing step, say
* TOLA = max(M,N)*norm(A)*MAZHEPS,
* TOLB = max(P,N)*norm(B)*MAZHEPS.
*
* ALPHA (output) DOUBLE PRECISION array, dimension (N)
* BETA (output) DOUBLE PRECISION array, dimension (N)
* On exit, ALPHA and BETA contain the generalized singular
* value pairs of A and B;
* ALPHA(1:K) = 1,
* BETA(1:K) = 0,
* and if M-K-L >= 0,
* ALPHA(K+1:K+L) = diag(C),
* BETA(K+1:K+L) = diag(S),
* or if M-K-L < 0,
* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
* BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
* Furthermore, if K+L < N,
* ALPHA(K+L+1:N) = 0 and
* BETA(K+L+1:N) = 0.
*
* U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
* On entry, if JOBU = 'U', U must contain a matrix U1 (usually
* the orthogonal matrix returned by DGGSVP).
* On exit,
* if JOBU = 'I', U contains the orthogonal matrix U;
* if JOBU = 'U', U contains the product U1*U.
* If JOBU = 'N', U is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= max(1,M) if
* JOBU = 'U'; LDU >= 1 otherwise.
*
* V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
* On entry, if JOBV = 'V', V must contain a matrix V1 (usually
* the orthogonal matrix returned by DGGSVP).
* On exit,
* if JOBV = 'I', V contains the orthogonal matrix V;
* if JOBV = 'V', V contains the product V1*V.
* If JOBV = 'N', V is not referenced.
*
* LDV (input) INTEGER
* The leading dimension of the array V. LDV >= max(1,P) if
* JOBV = 'V'; LDV >= 1 otherwise.
*
* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
* the orthogonal matrix returned by DGGSVP).
* On exit,
* if JOBQ = 'I', Q contains the orthogonal matrix Q;
* if JOBQ = 'Q', Q contains the product Q1*Q.
* If JOBQ = 'N', Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= max(1,N) if
* JOBQ = 'Q'; LDQ >= 1 otherwise.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
*
* NCYCLE (output) INTEGER
* The number of cycles required for convergence.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* = 1: the procedure does not converge after MAXIT cycles.
*
* Internal Parameters
* ===================
*
* MAXIT INTEGER
* MAXIT specifies the total loops that the iterative procedure
* may take. If after MAXIT cycles, the routine fails to
* converge, we return INFO = 1.
*
* Further Details
* ===============
*
* DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
* matrix B13 to the form:
*
* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
*
* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
* of Z. C1 and S1 are diagonal matrices satisfying
*
* C1**2 + S1**2 = I,
*
* and R1 is an L-by-L nonsingular upper triangular matrix.
*
* =====================================================================
*
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dtgsna
USAGE:
s, dif, m, work, info = NumRu::Lapack.dtgsna( job, howmny, select, a, b, vl, vr, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO )
* Purpose
* =======
*
* DTGSNA estimates reciprocal condition numbers for specified
* eigenvalues and/or eigenvectors of a matrix pair (A, B) in
* generalized real Schur canonical form (or of any matrix pair
* (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where
* Z' denotes the transpose of Z.
*
* (A, B) must be in generalized real Schur 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.
*
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies whether condition numbers are required for
* eigenvalues (S) or eigenvectors (DIF):
* = 'E': for eigenvalues only (S);
* = 'V': for eigenvectors only (DIF);
* = 'B': for both eigenvalues and eigenvectors (S and DIF).
*
* HOWMNY (input) CHARACTER*1
* = 'A': compute condition numbers for all eigenpairs;
* = 'S': compute condition numbers for selected eigenpairs
* specified by the array SELECT.
*
* SELECT (input) LOGICAL array, dimension (N)
* If HOWMNY = 'S', SELECT specifies the eigenpairs for which
* condition numbers are required. To select condition numbers
* for the eigenpair corresponding to a real eigenvalue w(j),
* SELECT(j) must be set to .TRUE.. To select condition numbers
* corresponding to a complex conjugate pair of eigenvalues w(j)
* and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
* set to .TRUE..
* If HOWMNY = 'A', SELECT is not referenced.
*
* N (input) INTEGER
* The order of the square matrix pair (A, B). N >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The upper quasi-triangular matrix A in the pair (A,B).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input) DOUBLE PRECISION array, dimension (LDB,N)
* The upper triangular matrix B in the pair (A,B).
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* VL (input) DOUBLE PRECISION array, dimension (LDVL,M)
* If JOB = 'E' or 'B', VL must contain left eigenvectors of
* (A, B), corresponding to the eigenpairs specified by HOWMNY
* and SELECT. The eigenvectors must be stored in consecutive
* columns of VL, as returned by DTGEVC.
* If JOB = 'V', VL is not referenced.
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1.
* If JOB = 'E' or 'B', LDVL >= N.
*
* VR (input) DOUBLE PRECISION array, dimension (LDVR,M)
* If JOB = 'E' or 'B', VR must contain right eigenvectors of
* (A, B), corresponding to the eigenpairs specified by HOWMNY
* and SELECT. The eigenvectors must be stored in consecutive
* columns ov VR, as returned by DTGEVC.
* If JOB = 'V', VR is not referenced.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1.
* If JOB = 'E' or 'B', LDVR >= N.
*
* S (output) DOUBLE PRECISION array, dimension (MM)
* If JOB = 'E' or 'B', the reciprocal condition numbers of the
* selected eigenvalues, stored in consecutive elements of the
* array. For a complex conjugate pair of eigenvalues two
* consecutive elements of S are set to the same value. Thus
* S(j), DIF(j), and the j-th columns of VL and VR all
* correspond to the same eigenpair (but not in general the
* j-th eigenpair, unless all eigenpairs are selected).
* If JOB = 'V', S is not referenced.
*
* DIF (output) DOUBLE PRECISION array, dimension (MM)
* If JOB = 'V' or 'B', the estimated reciprocal condition
* numbers of the selected eigenvectors, stored in consecutive
* elements of the array. For a complex eigenvector two
* consecutive elements of DIF are set to the same value. If
* the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
* is set to 0; this can only occur when the true value would be
* very small anyway.
* If JOB = 'E', DIF is not referenced.
*
* MM (input) INTEGER
* The number of elements in the arrays S and DIF. MM >= M.
*
* M (output) INTEGER
* The number of elements of the arrays S and DIF used to store
* the specified condition numbers; for each selected real
* eigenvalue one element is used, and for each selected complex
* conjugate pair of eigenvalues, two elements are used.
* If HOWMNY = 'A', M is set to N.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
* If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+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.
*
* IWORK (workspace) INTEGER array, dimension (N + 6)
* If JOB = 'E', IWORK is not referenced.
*
* INFO (output) INTEGER
* =0: Successful exit
* <0: If INFO = -i, the i-th argument had an illegal value
*
*
* Further Details
* ===============
*
* The reciprocal of the condition number of a generalized eigenvalue
* w = (a, b) is defined as
*
* S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))
*
* where u and v are the left and right eigenvectors of (A, B)
* corresponding to w; |z| denotes the absolute value of the complex
* number, and norm(u) denotes the 2-norm of the vector u.
* The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv)
* of the matrix pair (A, B). If both a and b equal zero, then (A B) is
* singular and S(I) = -1 is returned.
*
* An approximate error bound on the chordal distance between the i-th
* computed generalized eigenvalue w and the corresponding exact
* eigenvalue lambda is
*
* chord(w, lambda) <= EPS * norm(A, B) / S(I)
*
* where EPS is the machine precision.
*
* The reciprocal of the condition number DIF(i) of right eigenvector u
* and left eigenvector v corresponding to the generalized eigenvalue w
* is defined as follows:
*
* a) If the i-th eigenvalue w = (a,b) is real
*
* Suppose U and V are orthogonal transformations such that
*
* U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
* ( 0 S22 ),( 0 T22 ) n-1
* 1 n-1 1 n-1
*
* Then the reciprocal condition number DIF(i) is
*
* Difl((a, b), (S22, T22)) = sigma-min( Zl ),
*
* where sigma-min(Zl) denotes the smallest singular value of the
* 2(n-1)-by-2(n-1) matrix
*
* Zl = [ kron(a, In-1) -kron(1, S22) ]
* [ kron(b, In-1) -kron(1, T22) ] .
*
* Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
* Kronecker product between the matrices X and Y.
*
* Note that if the default method for computing DIF(i) is wanted
* (see DLATDF), then the parameter DIFDRI (see below) should be
* changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
* See DTGSYL for more details.
*
* b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
*
* Suppose U and V are orthogonal transformations such that
*
* U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
* ( 0 S22 ),( 0 T22) n-2
* 2 n-2 2 n-2
*
* and (S11, T11) corresponds to the complex conjugate eigenvalue
* pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
* that
*
* U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 )
* ( 0 s22 ) ( 0 t22 )
*
* where the generalized eigenvalues w = s11/t11 and
* conjg(w) = s22/t22.
*
* Then the reciprocal condition number DIF(i) is bounded by
*
* min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
*
* where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
* Z1 is the complex 2-by-2 matrix
*
* Z1 = [ s11 -s22 ]
* [ t11 -t22 ],
*
* This is done by computing (using real arithmetic) the
* roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
* where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
* the determinant of X.
*
* and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
* upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
*
* Z2 = [ kron(S11', In-2) -kron(I2, S22) ]
* [ kron(T11', In-2) -kron(I2, T22) ]
*
* Note that if the default method for computing DIF is wanted (see
* DLATDF), then the parameter DIFDRI (see below) should be changed
* from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
* for more details.
*
* For each eigenvalue/vector specified by SELECT, DIF stores a
* Frobenius norm-based estimate of Difl.
*
* An approximate error bound for the i-th computed eigenvector VL(i) or
* VR(i) is given by
*
* EPS * norm(A, B) / DIF(i).
*
* See ref. [2-3] for more details and further references.
*
* Based on contributions by
* 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.
*
* [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.
*
* =====================================================================
*
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dtgsy2
USAGE:
scale, pq, info, c, f, rdsum, rdscal = NumRu::Lapack.dtgsy2( trans, ijob, a, b, c, d, e, f, rdsum, rdscal, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO )
* Purpose
* =======
*
* DTGSY2 solves the generalized Sylvester equation:
*
* A * R - L * B = scale * C (1)
* D * R - L * E = scale * F,
*
* using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
* (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
* N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
* must be in generalized Schur canonical form, i.e. A, B are upper
* quasi triangular and D, E are upper triangular. The solution (R, L)
* overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
* chosen to avoid overflow.
*
* In matrix notation solving equation (1) corresponds to solve
* Z*x = scale*b, where Z is defined as
*
* Z = [ kron(In, A) -kron(B', Im) ] (2)
* [ kron(In, D) -kron(E', Im) ],
*
* Ik is the identity matrix of size k and X' is the transpose of X.
* kron(X, Y) is the Kronecker product between the matrices X and Y.
* In the process of solving (1), we solve a number of such systems
* where Dim(In), Dim(In) = 1 or 2.
*
* If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,
* which is equivalent to solve for R and L in
*
* A' * R + D' * L = scale * C (3)
* R * B' + L * E' = scale * -F
*
* This case is used to compute an estimate of Dif[(A, D), (B, E)] =
* sigma_min(Z) using reverse communicaton with DLACON.
*
* DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
* of an upper bound on the separation between to matrix pairs. Then
* the input (A, D), (B, E) are sub-pencils of the matrix pair in
* DTGSYL. See DTGSYL for details.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* = 'N', solve the generalized Sylvester equation (1).
* = 'T': solve the 'transposed' system (3).
*
* IJOB (input) INTEGER
* Specifies what kind of functionality to be performed.
* = 0: solve (1) only.
* = 1: A contribution from this subsystem to a Frobenius
* norm-based estimate of the separation between two matrix
* pairs is computed. (look ahead strategy is used).
* = 2: A contribution from this subsystem to a Frobenius
* norm-based estimate of the separation between two matrix
* pairs is computed. (DGECON on sub-systems is used.)
* Not referenced if TRANS = 'T'.
*
* M (input) INTEGER
* On entry, M specifies the order of A and D, and the row
* dimension of C, F, R and L.
*
* N (input) INTEGER
* On entry, N specifies the order of B and E, and the column
* dimension of C, F, R and L.
*
* A (input) DOUBLE PRECISION array, dimension (LDA, M)
* On entry, A contains an upper quasi triangular matrix.
*
* LDA (input) INTEGER
* The leading dimension of the matrix A. LDA >= max(1, M).
*
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, B contains an upper quasi triangular matrix.
*
* LDB (input) INTEGER
* The leading dimension of the matrix B. LDB >= max(1, N).
*
* C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
* On entry, C contains the right-hand-side of the first matrix
* equation in (1).
* On exit, if IJOB = 0, C has been overwritten by the
* solution R.
*
* LDC (input) INTEGER
* The leading dimension of the matrix C. LDC >= max(1, M).
*
* D (input) DOUBLE PRECISION array, dimension (LDD, M)
* On entry, D contains an upper triangular matrix.
*
* LDD (input) INTEGER
* The leading dimension of the matrix D. LDD >= max(1, M).
*
* E (input) DOUBLE PRECISION array, dimension (LDE, N)
* On entry, E contains an upper triangular matrix.
*
* LDE (input) INTEGER
* The leading dimension of the matrix E. LDE >= max(1, N).
*
* F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
* On entry, F contains the right-hand-side of the second matrix
* equation in (1).
* On exit, if IJOB = 0, F has been overwritten by the
* solution L.
*
* LDF (input) INTEGER
* The leading dimension of the matrix F. LDF >= max(1, M).
*
* SCALE (output) DOUBLE PRECISION
* On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
* R and L (C and F on entry) will hold the solutions to a
* slightly perturbed system but the input matrices A, B, D and
* E have not been changed. If SCALE = 0, R and L will hold the
* solutions to the homogeneous system with C = F = 0. Normally,
* SCALE = 1.
*
* RDSUM (input/output) DOUBLE PRECISION
* On entry, the sum of squares of computed contributions to
* the Dif-estimate under computation by DTGSYL, where the
* scaling factor RDSCAL (see below) has been factored out.
* On exit, the corresponding sum of squares updated with the
* contributions from the current sub-system.
* If TRANS = 'T' RDSUM is not touched.
* NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.
*
* RDSCAL (input/output) DOUBLE PRECISION
* On entry, scaling factor used to prevent overflow in RDSUM.
* On exit, RDSCAL is updated w.r.t. the current contributions
* in RDSUM.
* If TRANS = 'T', RDSCAL is not touched.
* NOTE: RDSCAL only makes sense when DTGSY2 is called by
* DTGSYL.
*
* IWORK (workspace) INTEGER array, dimension (M+N+2)
*
* PQ (output) INTEGER
* On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
* 8-by-8) solved by this routine.
*
* INFO (output) INTEGER
* On exit, if INFO is set to
* =0: Successful exit
* <0: If INFO = -i, the i-th argument had an illegal value.
* >0: The matrix pairs (A, D) and (B, E) have common or very
* close eigenvalues.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 27/5/02.
*
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dtgsyl
USAGE:
scale, dif, work, info, c, f = NumRu::Lapack.dtgsyl( trans, ijob, a, b, c, d, e, f, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO )
* Purpose
* =======
*
* DTGSYL solves the generalized Sylvester equation:
*
* A * R - L * B = scale * C (1)
* D * R - L * E = scale * F
*
* where R and L are unknown m-by-n matrices, (A, D), (B, E) and
* (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
* respectively, with real entries. (A, D) and (B, E) must be in
* generalized (real) Schur canonical form, i.e. A, B are upper quasi
* triangular and D, E are upper triangular.
*
* The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
* scaling factor chosen to avoid overflow.
*
* In matrix notation (1) is equivalent to solve Zx = scale b, where
* Z is defined as
*
* Z = [ kron(In, A) -kron(B', Im) ] (2)
* [ kron(In, D) -kron(E', Im) ].
*
* Here Ik is the identity matrix of size k and X' is the transpose of
* X. kron(X, Y) is the Kronecker product between the matrices X and Y.
*
* If TRANS = 'T', DTGSYL solves the transposed system Z'*y = scale*b,
* which is equivalent to solve for R and L in
*
* A' * R + D' * L = scale * C (3)
* R * B' + L * E' = scale * (-F)
*
* This case (TRANS = 'T') is used to compute an one-norm-based estimate
* of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
* and (B,E), using DLACON.
*
* If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
* of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
* reciprocal of the smallest singular value of Z. See [1-2] for more
* information.
*
* This is a level 3 BLAS algorithm.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* = 'N', solve the generalized Sylvester equation (1).
* = 'T', solve the 'transposed' system (3).
*
* IJOB (input) INTEGER
* Specifies what kind of functionality to be performed.
* =0: solve (1) only.
* =1: The functionality of 0 and 3.
* =2: The functionality of 0 and 4.
* =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
* (look ahead strategy IJOB = 1 is used).
* =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
* ( DGECON on sub-systems is used ).
* Not referenced if TRANS = 'T'.
*
* M (input) INTEGER
* The order of the matrices A and D, and the row dimension of
* the matrices C, F, R and L.
*
* N (input) INTEGER
* The order of the matrices B and E, and the column dimension
* of the matrices C, F, R and L.
*
* A (input) DOUBLE PRECISION array, dimension (LDA, M)
* The upper quasi triangular matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1, M).
*
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
* The upper quasi triangular matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1, N).
*
* C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
* On entry, C contains the right-hand-side of the first matrix
* equation in (1) or (3).
* On exit, if IJOB = 0, 1 or 2, C has been overwritten by
* the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
* the solution achieved during the computation of the
* Dif-estimate.
*
* LDC (input) INTEGER
* The leading dimension of the array C. LDC >= max(1, M).
*
* D (input) DOUBLE PRECISION array, dimension (LDD, M)
* The upper triangular matrix D.
*
* LDD (input) INTEGER
* The leading dimension of the array D. LDD >= max(1, M).
*
* E (input) DOUBLE PRECISION array, dimension (LDE, N)
* The upper triangular matrix E.
*
* LDE (input) INTEGER
* The leading dimension of the array E. LDE >= max(1, N).
*
* F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
* On entry, F contains the right-hand-side of the second matrix
* equation in (1) or (3).
* On exit, if IJOB = 0, 1 or 2, F has been overwritten by
* the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
* the solution achieved during the computation of the
* Dif-estimate.
*
* LDF (input) INTEGER
* The leading dimension of the array F. LDF >= max(1, M).
*
* DIF (output) DOUBLE PRECISION
* On exit DIF is the reciprocal of a lower bound of the
* reciprocal of the Dif-function, i.e. DIF is an upper bound of
* Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
* IF IJOB = 0 or TRANS = 'T', DIF is not touched.
*
* SCALE (output) DOUBLE PRECISION
* On exit SCALE is the scaling factor in (1) or (3).
* If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
* to a slightly perturbed system but the input matrices A, B, D
* and E have not been changed. If SCALE = 0, C and F hold the
* solutions R and L, respectively, to the homogeneous system
* with C = F = 0. Normally, SCALE = 1.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK > = 1.
* If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
*
* 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.
*
* IWORK (workspace) INTEGER array, dimension (M+N+6)
*
* INFO (output) INTEGER
* =0: successful exit
* <0: If INFO = -i, the i-th argument had an illegal value.
* >0: (A, D) and (B, E) have common or close eigenvalues.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* [1] 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.
*
* [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
* Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
* Appl., 15(4):1045-1060, 1994
*
* [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
* Condition Estimators for Solving the Generalized Sylvester
* Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
* July 1989, pp 745-751.
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 1/5/02.
*
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