## table of contents

complexGEeigen(3) | LAPACK | complexGEeigen(3) |

# NAME¶

complexGEeigen# SYNOPSIS¶

## Functions¶

subroutine

**cgees**(JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO)

**CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**subroutine

**cgeesx**(JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO)

**CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**subroutine

**cgeev**(JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

**CGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**subroutine

**cgeevx**(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, INFO)

**CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**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

**cgges3**(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)

**CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)**subroutine

**cggesx**(JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)

**CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**subroutine

**cggev**(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

**CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**subroutine

**cggev3**(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

**CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)**subroutine

**cggevx**(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO)

**CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**

# Detailed Description¶

This is the group of complex eigenvalue driver functions for GE matrices# Function Documentation¶

## subroutine cgees (character JOBVS, character SORT, external SELECT, integer N, complex, dimension( lda, * ) A, integer LDA, integer SDIM, complex, dimension( * ) W, complex, dimension( ldvs, * ) VS, integer LDVS, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)¶

**CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**

**Purpose: **

CGEES computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of Z then form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues. A complex matrix is in Schur form if it is upper triangular.

**Parameters:**

*JOBVS*

JOBVS is CHARACTER*1 = 'N': Schur vectors are not computed; = 'V': Schur vectors are computed.

*SORT*

SORT is CHARACTER*1 Specifies whether or not to order the eigenvalues on the diagonal of the Schur form. = 'N': Eigenvalues are not ordered: = 'S': Eigenvalues are ordered (see SELECT).

*SELECT*

SELECT is a LOGICAL FUNCTION of one COMPLEX argument SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT is used to select eigenvalues to order to the top left of the Schur form. IF SORT = 'N', SELECT is not referenced. The eigenvalue W(j) is selected if SELECT(W(j)) is true.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX array, dimension (LDA,N) On entry, the N-by-N matrix A. On exit, A has been overwritten by its Schur form T.

*LDA*

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

*SDIM*

SDIM is INTEGER If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues for which SELECT is true.

*W*

W is COMPLEX array, dimension (N) W contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.

*VS*

VS is COMPLEX array, dimension (LDVS,N) If JOBVS = 'V', VS contains the unitary matrix Z of Schur vectors. If JOBVS = 'N', VS is not referenced.

*LDVS*

LDVS is INTEGER The leading dimension of the array VS. LDVS >= 1; if JOBVS = 'V', LDVS >= 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 (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. > 0: if INFO = i, and i is <= N: the QR algorithm failed to compute all the eigenvalues; elements 1:ILO-1 and i+1:N of W contain those eigenvalues which have converged; if JOBVS = 'V', VS contains the matrix which reduces A to its partially converged Schur form. = N+1: the eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned); = N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy SELECT = .TRUE.. This could also be caused by underflow due to scaling.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer N, complex, dimension( lda, * ) A, integer LDA, integer SDIM, complex, dimension( * ) W, complex, dimension( ldvs, * ) VS, integer LDVS, real RCONDE, real RCONDV, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)¶

**CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**

**Purpose: **

CGEESX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (RCONDV). The leading columns of Z form an orthonormal basis for this invariant subspace. For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where these quantities are called s and sep respectively). A complex matrix is in Schur form if it is upper triangular.

**Parameters:**

*JOBVS*

JOBVS is CHARACTER*1 = 'N': Schur vectors are not computed; = 'V': Schur vectors are computed.

*SORT*

SORT is CHARACTER*1 Specifies whether or not to order the eigenvalues on the diagonal of the Schur form. = 'N': Eigenvalues are not ordered; = 'S': Eigenvalues are ordered (see SELECT).

*SELECT*

SELECT is a LOGICAL FUNCTION of one COMPLEX argument SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT is used to select eigenvalues to order to the top left of the Schur form. If SORT = 'N', SELECT is not referenced. An eigenvalue W(j) is selected if SELECT(W(j)) is true.

*SENSE*

SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': None are computed; = 'E': Computed for average of selected eigenvalues only; = 'V': Computed for selected right invariant subspace only; = 'B': Computed for both. If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX array, dimension (LDA, N) On entry, the N-by-N matrix A. On exit, A is overwritten by its Schur form T.

*LDA*

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

*SDIM*

SDIM is INTEGER If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues for which SELECT is true.

*W*

W is COMPLEX array, dimension (N) W contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.

*VS*

VS is COMPLEX array, dimension (LDVS,N) If JOBVS = 'V', VS contains the unitary matrix Z of Schur vectors. If JOBVS = 'N', VS is not referenced.

*LDVS*

LDVS is INTEGER The leading dimension of the array VS. LDVS >= 1, and if JOBVS = 'V', LDVS >= N.

*RCONDE*

RCONDE is REAL If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition number for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.

*RCONDV*

RCONDV is REAL If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition number for the selected right invariant subspace. Not referenced if SENSE = 'N' or 'E'.

*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). Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), where SDIM is the number of selected eigenvalues computed by this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < max(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough. For good performance, LWORK must generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates upper bound on the optimal size of the array WORK, 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 (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. > 0: if INFO = i, and i is <= N: the QR algorithm failed to compute all the eigenvalues; elements 1:ILO-1 and i+1:N of W contain those eigenvalues which have converged; if JOBVS = 'V', VS contains the transformation which reduces A to its partially converged Schur form. = N+1: the eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned); = N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy SELECT=.TRUE. This could also be caused by underflow due to scaling.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cgeev (character JOBVL, character JOBVR, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) W, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)¶

**CGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**

**Purpose: **

CGEEV computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate transpose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

**Parameters:**

*JOBVL*

JOBVL is CHARACTER*1 = 'N': left eigenvectors of A are not computed; = 'V': left eigenvectors of are computed.

*JOBVR*

JOBVR is CHARACTER*1 = 'N': right eigenvectors of A are not computed; = 'V': right eigenvectors of A are computed.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX array, dimension (LDA,N) On entry, the N-by-N matrix A. On exit, A has been overwritten.

*LDA*

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

*W*

W is COMPLEX array, dimension (N) W contains the computed eigenvalues.

*VL*

VL is COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If JOBVL = 'N', VL is not referenced. u(j) = VL(:,j), the j-th column of VL.

*LDVL*

LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.

*VR*

VR is COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If JOBVR = 'N', VR is not referenced. v(j) = VR(:,j), the j-th column of VR.

*LDVR*

LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= 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 (2*N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements and i+1:N of W contain eigenvalues which have converged.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cgeevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) W, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, real, dimension( * ) SCALE, real ABNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)¶

**CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**

**Purpose: **

CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate transpose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation D * A * D**(-1), where D is a diagonal matrix, to make its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.10.2 of the LAPACK Users' Guide.

**Parameters:**

*BALANC*

BALANC is CHARACTER*1 Indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues. = 'N': Do not diagonally scale or permute; = 'P': Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale; = 'S': Diagonally scale the matrix, ie. replace A by D*A*D**(-1), where D is a diagonal matrix chosen to make the rows and columns of A more equal in norm. Do not permute; = 'B': Both diagonally scale and permute A. Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

*JOBVL*

JOBVL is CHARACTER*1 = 'N': left eigenvectors of A are not computed; = 'V': left eigenvectors of A are computed. If SENSE = 'E' or 'B', JOBVL must = 'V'.

*JOBVR*

JOBVR is CHARACTER*1 = 'N': right eigenvectors of A are not computed; = 'V': right eigenvectors of A are computed. If SENSE = 'E' or 'B', JOBVR must = 'V'.

*SENSE*

SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': None are computed; = 'E': Computed for eigenvalues only; = 'V': Computed for right eigenvectors only; = 'B': Computed for eigenvalues and right eigenvectors. If SENSE = 'E' or 'B', both left and right eigenvectors must also be computed (JOBVL = 'V' and JOBVR = 'V').

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX array, dimension (LDA,N) On entry, the N-by-N matrix A. On exit, A has been overwritten. If JOBVL = 'V' or JOBVR = 'V', A contains the Schur form of the balanced version of the matrix A.

*LDA*

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

*W*

W is COMPLEX array, dimension (N) W contains the computed eigenvalues.

*VL*

VL is COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If JOBVL = 'N', VL is not referenced. u(j) = VL(:,j), the j-th column of VL.

*LDVL*

LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.

*VR*

VR is COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If JOBVR = 'N', VR is not referenced. v(j) = VR(:,j), the j-th column of VR.

*LDVR*

LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER ILO and IHI are integer values determined when A was balanced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO-1 or I = IHI+1,...,N.

*SCALE*

SCALE is REAL array, dimension (N) Details of the permutations and scaling factors applied when balancing A. If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and column j, then SCALE(J) = P(J), for J = 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J) for J = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

*ABNRM*

ABNRM is REAL The one-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).

*RCONDE*

RCONDE is REAL array, dimension (N) RCONDE(j) is the reciprocal condition number of the j-th eigenvalue.

*RCONDV*

RCONDV is REAL array, dimension (N) RCONDV(j) is the reciprocal condition number of the j-th right eigenvector.

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. If SENSE = 'N' or 'E', LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', LWORK >= N*N+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 (2*N)

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements 1:ILO-1 and i+1:N of W contain eigenvalues which have converged.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## 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*

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cgges3 (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)¶

**CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)**

**Purpose: **

CGGES3 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 CGGES3. 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*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cggesx (character JOBVSL, character JOBVSR, character SORT, external SELCTG, character SENSE, 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, real, dimension( 2 ) RCONDE, real, dimension( 2 ) RCONDV, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)¶

**CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices**

**Purpose: **

CGGESX computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of 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; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). 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 for both being zero. A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular.

**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. 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+3 see INFO below).

*SENSE*

SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N' : None are computed; = 'E' : Computed for average of selected eigenvalues only; = 'V' : Computed for selected deflating subspaces only; = 'B' : Computed for both. If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

*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) and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T). 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.

*RCONDE*

RCONDE is REAL array, dimension ( 2 ) If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.

*RCONDV*

RCONDV is REAL array, dimension ( 2 ) If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition number for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough. If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is REAL array, dimension ( 8*N ) Real workspace.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

*LIWORK*

LIWORK is INTEGER The dimension of the array WORK. If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise LIWORK >= N+2. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cggev (character JOBVL, character JOBVR, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)¶

**CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**

**Purpose: **

CGGEV computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*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. The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j). The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B where u(j)**H is the conjugate-transpose of u(j).

**Parameters:**

*JOBVL*

JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors.

*JOBVR*

JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors.

*N*

N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0.

*A*

A is COMPLEX array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten.

*LDA*

LDA is INTEGER The leading dimension of A. LDA >= max(1,N).

*B*

B is COMPLEX array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten.

*LDB*

LDB is INTEGER The leading dimension of B. LDB >= max(1,N).

*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. 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).

*VL*

VL is COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.

*LDVL*

LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.

*VR*

VR is COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.

*LDVR*

LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.

*WORK*

*LWORK*

*RWORK*

RWORK is REAL array, dimension (8*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. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other then QZ iteration failed in SHGEQZ, =N+2: error return from STGEVC.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cggev3 (character JOBVL, character JOBVR, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)¶

**CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)**

**Purpose: **

CGGEV3 computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*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. The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j). The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B where u(j)**H is the conjugate-transpose of u(j).

**Parameters:**

*JOBVL*

JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors.

*JOBVR*

JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors.

*N*

N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0.

*A*

A is COMPLEX array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten.

*LDA*

LDA is INTEGER The leading dimension of A. LDA >= max(1,N).

*B*

B is COMPLEX array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten.

*LDB*

LDB is INTEGER The leading dimension of B. LDB >= max(1,N).

*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. 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).

*VL*

VL is COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.

*LDVL*

LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.

*VR*

VR is COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.

*LDVR*

LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. 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)

*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. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other then QZ iteration failed in SHGEQZ, =N+2: error return from STGEVC.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine cggevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, real ABNRM, real BBNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)¶

**CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices**

**Purpose: **

CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*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. The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j) . The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B. where u(j)**H is the conjugate-transpose of u(j).

**Parameters:**

*BALANC*

BALANC is CHARACTER*1 Specifies the balance option to be performed: = 'N': do not diagonally scale or permute; = 'P': permute only; = 'S': scale only; = 'B': both permute and scale. Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

*JOBVL*

JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors.

*JOBVR*

JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors.

*SENSE*

SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': none are computed; = 'E': computed for eigenvalues only; = 'V': computed for eigenvectors only; = 'B': computed for eigenvalues and eigenvectors.

*N*

N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0.

*A*

A is COMPLEX array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the "balanced" versions of the input A and B.

*LDA*

LDA is INTEGER The leading dimension of A. LDA >= max(1,N).

*B*

B is COMPLEX array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the "balanced" versions of the input A and B.

*LDB*

LDB is INTEGER The leading dimension of B. LDB >= max(1,N).

*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. Note: the quotient 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).

*VL*

VL is COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.

*LDVL*

LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.

*VR*

VR is COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.

*LDVR*

LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

*LSCALE*

LSCALE is REAL array, dimension (N) Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

*RSCALE*

RSCALE is REAL array, dimension (N) Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then RSCALE(j) = PR(j) for j = 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j = IHI+1,...,N The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

*ABNRM*

ABNRM is REAL The one-norm of the balanced matrix A.

*BBNRM*

BBNRM is REAL The one-norm of the balanced matrix B.

*RCONDE*

RCONDE is REAL array, dimension (N) If SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. If SENSE = 'N' or 'V', RCONDE is not referenced.

*RCONDV*

RCONDV is REAL array, dimension (N) If SENSE = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'N' or 'E', RCONDV is not referenced.

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). If SENSE = 'E', LWORK >= max(1,4*N). If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*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.

*RWORK*

RWORK is REAL array, dimension (lrwork) lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and at least max(1,2*N) otherwise. Real workspace.

*IWORK*

IWORK is INTEGER array, dimension (N+2) If SENSE = 'E', IWORK is not referenced.

*BWORK*

BWORK is LOGICAL array, dimension (N) If SENSE = 'N', BWORK is not referenced.

*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. No eigenvectors have been calculated, 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: error return from CTGEVC.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide. 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(ABNRM, BBNRM) / RCONDE(I) An approximate error bound for the angle between the i-th computed eigenvector VL(i) or VR(i) is given by EPS * norm(ABNRM, BBNRM) / DIF(i). For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide.

# Author¶

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