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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine sgges3 | ( | character | jobvsl, |
| character | jobvsr, | ||
| character | sort, | ||
| external | selctg, | ||
| integer | n, | ||
| real, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| real, dimension( ldb, * ) | b, | ||
| integer | ldb, | ||
| integer | sdim, | ||
| real, dimension( * ) | alphar, | ||
| real, dimension( * ) | alphai, | ||
| real, dimension( * ) | beta, | ||
| real, dimension( ldvsl, * ) | vsl, | ||
| integer | ldvsl, | ||
| real, dimension( ldvsr, * ) | vsr, | ||
| integer | ldvsr, | ||
| real, dimension( * ) | work, | ||
| integer | lwork, | ||
| logical, dimension( * ) | bwork, | ||
| integer | info ) |
SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
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!> !> SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B), !> the generalized eigenvalues, the generalized real Schur form (S,T), !> optionally, the left and/or right matrices of Schur vectors (VSL and !> VSR). This gives the generalized Schur factorization !> !> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) !> !> Optionally, it also orders the eigenvalues so that a selected cluster !> of eigenvalues appears in the leading diagonal blocks of the upper !> quasi-triangular matrix S and the upper triangular matrix T.The !> leading columns of VSL and VSR then form an orthonormal basis for the !> corresponding left and right eigenspaces (deflating subspaces). !> !> (If only the generalized eigenvalues are needed, use the driver !> SGGEV instead, which is faster.) !> !> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w !> or a ratio alpha/beta = w, such that A - w*B is singular. It is !> usually represented as the pair (alpha,beta), as there is a !> reasonable interpretation for beta=0 or both being zero. !> !> A pair of matrices (S,T) is in generalized real Schur form if T is !> upper triangular with non-negative diagonal and S is block upper !> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond !> to real generalized eigenvalues, while 2-by-2 blocks of S will be !> by making the corresponding elements of T have the !> form: !> [ a 0 ] !> [ 0 b ] !> !> and the pair of corresponding 2-by-2 blocks in S and T will have a !> complex conjugate pair of generalized eigenvalues. !> !>
| [in] | JOBVSL | !> JOBVSL is CHARACTER*1 !> = 'N': do not compute the left Schur vectors; !> = 'V': compute the left Schur vectors. !> |
| [in] | JOBVSR | !> JOBVSR is CHARACTER*1 !> = 'N': do not compute the right Schur vectors; !> = 'V': compute the right Schur vectors. !> |
| [in] | SORT | !> SORT is CHARACTER*1 !> Specifies whether or not to order the eigenvalues on the !> diagonal of the generalized Schur form. !> = 'N': Eigenvalues are not ordered; !> = 'S': Eigenvalues are ordered (see SELCTG); !> |
| [in] | SELCTG | !> SELCTG is a LOGICAL FUNCTION of three REAL arguments !> SELCTG must be declared EXTERNAL in the calling subroutine. !> If SORT = 'N', SELCTG is not referenced. !> If SORT = 'S', SELCTG is used to select eigenvalues to sort !> to the top left of the Schur form. !> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if !> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either !> one of a complex conjugate pair of eigenvalues is selected, !> then both complex eigenvalues are selected. !> !> Note that in the ill-conditioned case, a selected complex !> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), !> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 !> in this case. !> |
| [in] | N | !> N is INTEGER !> The order of the matrices A, B, VSL, and VSR. N >= 0. !> |
| [in,out] | A | !> A is REAL array, dimension (LDA, N) !> On entry, the first of the pair of matrices. !> On exit, A has been overwritten by its generalized Schur !> form S. !> |
| [in] | LDA | !> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !> |
| [in,out] | B | !> B is REAL array, dimension (LDB, N) !> On entry, the second of the pair of matrices. !> On exit, B has been overwritten by its generalized Schur !> form T. !> |
| [in] | LDB | !> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !> |
| [out] | SDIM | !> SDIM is INTEGER !> If SORT = 'N', SDIM = 0. !> If SORT = 'S', SDIM = number of eigenvalues (after sorting) !> for which SELCTG is true. (Complex conjugate pairs for which !> SELCTG is true for either eigenvalue count as 2.) !> |
| [out] | ALPHAR | !> ALPHAR is REAL array, dimension (N) !> |
| [out] | ALPHAI | !> ALPHAI is REAL array, dimension (N) !> |
| [out] | BETA | !> BETA is REAL 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 Schur form of (A,B) were further reduced to !> triangular form using 2-by-2 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. !> !> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) !> may easily over- or underflow, and BETA(j) may even be zero. !> Thus, the user should avoid naively computing the ratio. !> However, ALPHAR and ALPHAI will be always less than and !> usually comparable with norm(A) in magnitude, and BETA always !> less than and usually comparable with norm(B). !> |
| [out] | VSL | !> VSL is REAL array, dimension (LDVSL,N) !> If JOBVSL = 'V', VSL will contain the left Schur vectors. !> Not referenced if JOBVSL = 'N'. !> |
| [in] | LDVSL | !> LDVSL is INTEGER !> The leading dimension of the matrix VSL. LDVSL >=1, and !> if JOBVSL = 'V', LDVSL >= N. !> |
| [out] | VSR | !> VSR is REAL array, dimension (LDVSR,N) !> If JOBVSR = 'V', VSR will contain the right Schur vectors. !> Not referenced if JOBVSR = 'N'. !> |
| [in] | LDVSR | !> LDVSR is INTEGER !> The leading dimension of the matrix VSR. LDVSR >= 1, and !> if JOBVSR = 'V', LDVSR >= N. !> |
| [out] | WORK | !> WORK is REAL array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> |
| [in] | LWORK | !> LWORK is INTEGER !> The dimension of the array WORK. !> If N = 0, LWORK >= 1, else LWORK >= 6*N+16. !> For good performance, LWORK must generally be larger. !> !> 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. !> |
| [out] | BWORK | !> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !> |
| [out] | INFO | !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !> = 1,...,N: !> The QZ iteration failed. (A,B) are not in Schur !> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should !> be correct for j=INFO+1,...,N. !> > N: =N+1: other than QZ iteration failed in SLAQZ0. !> =N+2: after reordering, roundoff changed values of !> some complex eigenvalues so that leading !> eigenvalues in the Generalized Schur form no !> longer satisfy SELCTG=.TRUE. This could also !> be caused due to scaling. !> =N+3: reordering failed in STGSEN. !> |
Definition at line 279 of file sgges3.f.
1.11.0