--- name: "gges" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/gges" --- # gges(3) gges: Schur form, unblocked ## NAME gges - gges: Schur form, unblocked ## SYNOPSIS ### Functions subroutine **cgges** (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)\ **CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** subroutine **dgges** (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)\ **DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** subroutine **sgges** (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)\ **SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** subroutine **zgges** (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)\ **ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** ## Detailed Description ## Function Documentation ### subroutine cgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex, dimension( lda, \* ) a, integer lda, complex, dimension( ldb, \* ) b, integer ldb, integer sdim, complex, dimension( \* ) alpha, complex, dimension( \* ) beta, complex, dimension( ldvsl, \* ) vsl, integer ldvsl, complex, dimension( ldvsr, \* ) vsr, integer ldvsr, complex, dimension( \* ) work, integer lwork, real, dimension( \* ) rwork, logical, dimension( \* ) bwork, integer info) **CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** **Purpose:** ``` !> !> CGGES computes for a pair of N-by-N complex nonsymmetric matrices !> (A,B), the generalized eigenvalues, the generalized complex Schur !> form (S, T), and optionally left and/or right Schur vectors (VSL !> and VSR). This gives the generalized Schur factorization !> !> (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H ) !> !> where (VSR)**H is the conjugate-transpose of VSR. !> !> Optionally, it also orders the eigenvalues so that a selected cluster !> of eigenvalues appears in the leading diagonal blocks of the upper !> triangular matrix S and the upper triangular matrix T. The leading !> columns of VSL and VSR then form an unitary basis for the !> corresponding left and right eigenspaces (deflating subspaces). !> !> (If only the generalized eigenvalues are needed, use the driver !> CGGEV 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, and even for both being zero. !> !> A pair of matrices (S,T) is in generalized complex Schur form if S !> and T are upper triangular and, in addition, the diagonal elements !> of T are non-negative real numbers. !> ``` **Parameters** *JOBVSL* ``` !> JOBVSL is CHARACTER*1 !> = 'N': do not compute the left Schur vectors; !> = 'V': compute the left Schur vectors. !> ``` *JOBVSR* ``` !> JOBVSR is CHARACTER*1 !> = 'N': do not compute the right Schur vectors; !> = 'V': compute the right Schur vectors. !> ``` *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). !> ``` *SELCTG* ``` !> SELCTG is a LOGICAL FUNCTION of two COMPLEX 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 ALPHA(j)/BETA(j) is selected if !> SELCTG(ALPHA(j),BETA(j)) is true. !> !> Note that a selected complex eigenvalue may no longer satisfy !> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since !> ordering may change the value of complex eigenvalues !> (especially if the eigenvalue is ill-conditioned), in this !> case INFO is set to N+2 (See INFO below). !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, VSL, and VSR. N >= 0. !> ``` *A* ``` !> A is COMPLEX 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. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !> ``` *B* ``` !> B is COMPLEX 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. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !> ``` *SDIM* ``` !> SDIM is INTEGER !> If SORT = 'N', SDIM = 0. !> If SORT = 'S', SDIM = number of eigenvalues (after sorting) !> for which SELCTG is true. !> ``` *ALPHA* ``` !> ALPHA is COMPLEX array, dimension (N) !> ``` *BETA* ``` !> BETA is COMPLEX array, dimension (N) !> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the !> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), !> j=1,...,N are the diagonals of the complex Schur form (A,B) !> output by CGGES. The BETA(j) will be non-negative real. !> !> Note: the quotients ALPHA(j)/BETA(j) may easily over- or !> underflow, and BETA(j) may even be zero. Thus, the user !> should avoid naively computing the ratio alpha/beta. !> However, ALPHA will be always less than and usually !> comparable with norm(A) in magnitude, and BETA always less !> than and usually comparable with norm(B). !> ``` *VSL* ``` !> VSL is COMPLEX array, dimension (LDVSL,N) !> If JOBVSL = 'V', VSL will contain the left Schur vectors. !> Not referenced if JOBVSL = 'N'. !> ``` *LDVSL* ``` !> LDVSL is INTEGER !> The leading dimension of the matrix VSL. LDVSL >= 1, and !> if JOBVSL = 'V', LDVSL >= N. !> ``` *VSR* ``` !> VSR is COMPLEX array, dimension (LDVSR,N) !> If JOBVSR = 'V', VSR will contain the right Schur vectors. !> Not referenced if JOBVSR = 'N'. !> ``` *LDVSR* ``` !> LDVSR is INTEGER !> The leading dimension of the matrix VSR. LDVSR >= 1, and !> if JOBVSR = 'V', LDVSR >= N. !> ``` *WORK* ``` !> WORK is COMPLEX 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 >= max(1,2*N). !> 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. !> ``` *RWORK* ``` !> RWORK is REAL array, dimension (8*N) !> ``` *BWORK* ``` !> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !> ``` *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 ALPHA(j) and BETA(j) should be correct for !> j=INFO+1,...,N. !> > N: =N+1: other than QZ iteration failed in CHGEQZ !> =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 CTGSEN. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **267** of file **cgges.f**. ### subroutine dgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( ldb, \* ) b, integer ldb, integer sdim, double precision, dimension( \* ) alphar, double precision, dimension( \* ) alphai, double precision, dimension( \* ) beta, double precision, dimension( ldvsl, \* ) vsl, integer ldvsl, double precision, dimension( ldvsr, \* ) vsr, integer ldvsr, double precision, dimension( \* ) work, integer lwork, logical, dimension( \* ) bwork, integer info) **DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** **Purpose:** ``` !> !> DGGES 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 !> DGGEV 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. !> !> ``` **Parameters** *JOBVSL* ``` !> JOBVSL is CHARACTER*1 !> = 'N': do not compute the left Schur vectors; !> = 'V': compute the left Schur vectors. !> ``` *JOBVSR* ``` !> JOBVSR is CHARACTER*1 !> = 'N': do not compute the right Schur vectors; !> = 'V': compute the right Schur vectors. !> ``` *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); !> ``` *SELCTG* ``` !> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION 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. !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, VSL, and VSR. N >= 0. !> ``` *A* ``` !> A is DOUBLE PRECISION 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. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !> ``` *B* ``` !> B is DOUBLE PRECISION 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. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !> ``` *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.) !> ``` *ALPHAR* ``` !> ALPHAR is DOUBLE PRECISION array, dimension (N) !> ``` *ALPHAI* ``` !> ALPHAI is DOUBLE PRECISION array, dimension (N) !> ``` *BETA* ``` !> BETA is 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 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). !> ``` *VSL* ``` !> VSL is DOUBLE PRECISION array, dimension (LDVSL,N) !> If JOBVSL = 'V', VSL will contain the left Schur vectors. !> Not referenced if JOBVSL = 'N'. !> ``` *LDVSL* ``` !> LDVSL is INTEGER !> The leading dimension of the matrix VSL. LDVSL >=1, and !> if JOBVSL = 'V', LDVSL >= N. !> ``` *VSR* ``` !> VSR is DOUBLE PRECISION array, dimension (LDVSR,N) !> If JOBVSR = 'V', VSR will contain the right Schur vectors. !> Not referenced if JOBVSR = 'N'. !> ``` *LDVSR* ``` !> LDVSR is INTEGER !> The leading dimension of the matrix VSR. LDVSR >= 1, and !> if JOBVSR = 'V', LDVSR >= 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. !> If N = 0, LWORK >= 1, else LWORK >= MAX(8*N,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. !> ``` *BWORK* ``` !> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !> ``` *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 DHGEQZ. !> =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 DTGSEN. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **281** of file **dgges.f**. ### subroutine sgges (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) **SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** **Purpose:** ``` !> !> SGGES 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. !> !> ``` **Parameters** *JOBVSL* ``` !> JOBVSL is CHARACTER*1 !> = 'N': do not compute the left Schur vectors; !> = 'V': compute the left Schur vectors. !> ``` *JOBVSR* ``` !> JOBVSR is CHARACTER*1 !> = 'N': do not compute the right Schur vectors; !> = 'V': compute the right Schur vectors. !> ``` *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); !> ``` *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. !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, VSL, and VSR. N >= 0. !> ``` *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. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !> ``` *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. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !> ``` *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.) !> ``` *ALPHAR* ``` !> ALPHAR is REAL array, dimension (N) !> ``` *ALPHAI* ``` !> ALPHAI is REAL array, dimension (N) !> ``` *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). !> ``` *VSL* ``` !> VSL is REAL array, dimension (LDVSL,N) !> If JOBVSL = 'V', VSL will contain the left Schur vectors. !> Not referenced if JOBVSL = 'N'. !> ``` *LDVSL* ``` !> LDVSL is INTEGER !> The leading dimension of the matrix VSL. LDVSL >=1, and !> if JOBVSL = 'V', LDVSL >= N. !> ``` *VSR* ``` !> VSR is REAL array, dimension (LDVSR,N) !> If JOBVSR = 'V', VSR will contain the right Schur vectors. !> Not referenced if JOBVSR = 'N'. !> ``` *LDVSR* ``` !> LDVSR is INTEGER !> The leading dimension of the matrix VSR. LDVSR >= 1, and !> if JOBVSR = 'V', LDVSR >= 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. !> If N = 0, LWORK >= 1, else LWORK >= max(8*N,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. !> ``` *BWORK* ``` !> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !> ``` *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 SHGEQZ. !> =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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **281** of file **sgges.f**. ### subroutine zgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex\*16, dimension( lda, \* ) a, integer lda, complex\*16, dimension( ldb, \* ) b, integer ldb, integer sdim, complex\*16, dimension( \* ) alpha, complex\*16, dimension( \* ) beta, complex\*16, dimension( ldvsl, \* ) vsl, integer ldvsl, complex\*16, dimension( ldvsr, \* ) vsr, integer ldvsr, complex\*16, dimension( \* ) work, integer lwork, double precision, dimension( \* ) rwork, logical, dimension( \* ) bwork, integer info) **ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices** **Purpose:** ``` !> !> ZGGES computes for a pair of N-by-N complex nonsymmetric matrices !> (A,B), the generalized eigenvalues, the generalized complex Schur !> form (S, T), and optionally left and/or right Schur vectors (VSL !> and VSR). This gives the generalized Schur factorization !> !> (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H ) !> !> where (VSR)**H is the conjugate-transpose of VSR. !> !> Optionally, it also orders the eigenvalues so that a selected cluster !> of eigenvalues appears in the leading diagonal blocks of the upper !> triangular matrix S and the upper triangular matrix T. The leading !> columns of VSL and VSR then form an unitary basis for the !> corresponding left and right eigenspaces (deflating subspaces). !> !> (If only the generalized eigenvalues are needed, use the driver !> ZGGEV 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, and even for both being zero. !> !> A pair of matrices (S,T) is in generalized complex Schur form if S !> and T are upper triangular and, in addition, the diagonal elements !> of T are non-negative real numbers. !> ``` **Parameters** *JOBVSL* ``` !> JOBVSL is CHARACTER*1 !> = 'N': do not compute the left Schur vectors; !> = 'V': compute the left Schur vectors. !> ``` *JOBVSR* ``` !> JOBVSR is CHARACTER*1 !> = 'N': do not compute the right Schur vectors; !> = 'V': compute the right Schur vectors. !> ``` *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). !> ``` *SELCTG* ``` !> SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 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 ALPHA(j)/BETA(j) is selected if !> SELCTG(ALPHA(j),BETA(j)) is true. !> !> Note that a selected complex eigenvalue may no longer satisfy !> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since !> ordering may change the value of complex eigenvalues !> (especially if the eigenvalue is ill-conditioned), in this !> case INFO is set to N+2 (See INFO below). !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, VSL, and VSR. N >= 0. !> ``` *A* ``` !> A is COMPLEX*16 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. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of A. LDA >= max(1,N). !> ``` *B* ``` !> B is COMPLEX*16 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. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of B. LDB >= max(1,N). !> ``` *SDIM* ``` !> SDIM is INTEGER !> If SORT = 'N', SDIM = 0. !> If SORT = 'S', SDIM = number of eigenvalues (after sorting) !> for which SELCTG is true. !> ``` *ALPHA* ``` !> ALPHA is COMPLEX*16 array, dimension (N) !> ``` *BETA* ``` !> BETA is COMPLEX*16 array, dimension (N) !> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the !> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), !> j=1,...,N are the diagonals of the complex Schur form (A,B) !> output by ZGGES. The BETA(j) will be non-negative real. !> !> Note: the quotients ALPHA(j)/BETA(j) may easily over- or !> underflow, and BETA(j) may even be zero. Thus, the user !> should avoid naively computing the ratio alpha/beta. !> However, ALPHA will be always less than and usually !> comparable with norm(A) in magnitude, and BETA always less !> than and usually comparable with norm(B). !> ``` *VSL* ``` !> VSL is COMPLEX*16 array, dimension (LDVSL,N) !> If JOBVSL = 'V', VSL will contain the left Schur vectors. !> Not referenced if JOBVSL = 'N'. !> ``` *LDVSL* ``` !> LDVSL is INTEGER !> The leading dimension of the matrix VSL. LDVSL >= 1, and !> if JOBVSL = 'V', LDVSL >= N. !> ``` *VSR* ``` !> VSR is COMPLEX*16 array, dimension (LDVSR,N) !> If JOBVSR = 'V', VSR will contain the right Schur vectors. !> Not referenced if JOBVSR = 'N'. !> ``` *LDVSR* ``` !> LDVSR is INTEGER !> The leading dimension of the matrix VSR. LDVSR >= 1, and !> if JOBVSR = 'V', LDVSR >= N. !> ``` *WORK* ``` !> WORK is COMPLEX*16 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 >= max(1,2*N). !> 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. !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION array, dimension (8*N) !> ``` *BWORK* ``` !> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !> ``` *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 ALPHA(j) and BETA(j) should be correct for !> j=INFO+1,...,N. !> > N: =N+1: other than QZ iteration failed in ZHGEQZ !> =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 ZTGSEN. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **267** of file **zgges.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.