## table of contents

realGEeigen(3) | LAPACK | realGEeigen(3) |

# NAME¶

realGEeigen# SYNOPSIS¶

## Functions¶

subroutine

**sgees**(JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS, WORK, LWORK, BWORK, INFO)

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

**sgeesx**(JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)

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

**sgeev**(JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)

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

**sgeevx**(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)

**SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors 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

**sgges3**(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)

**SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)**subroutine

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

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

**sggev**(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)

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

**sggev3**(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)

**SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)**subroutine

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

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

# Detailed Description¶

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

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

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

**Purpose: **

SGEES computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). Optionally, it also orders the eigenvalues on the diagonal of the real 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 matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form [ a b ] [ c a ] where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

**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 two REAL arguments SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT is used to select eigenvalues to sort to the top left of the Schur form. If SORT = 'N', SELECT is not referenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if SELECT(WR(j),WI(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 a selected complex eigenvalue may no longer satisfy SELECT(WR(j),WI(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 matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N) On entry, the N-by-N matrix A. On exit, A has been overwritten by its real 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 (after sorting) for which SELECT is true. (Complex conjugate pairs for which SELECT is true for either eigenvalue count as 2.)

*WR*

WR is REAL array, dimension (N)

*WI*

WI is REAL array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

*VS*

VS is REAL array, dimension (LDVS,N) If JOBVS = 'V', VS contains the orthogonal 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 REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,3*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.

*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 WR and WI 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 sgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, integer SDIM, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvs, * ) VS, integer LDVS, real RCONDE, real RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)¶

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

**Purpose: **

SGEESX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). Optionally, it also orders the eigenvalues on the diagonal of the real 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 real matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form [ a b ] [ c a ] where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

**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 two REAL arguments SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT is used to select eigenvalues to sort to the top left of the Schur form. If SORT = 'N', SELECT is not referenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case INFO may be 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 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 REAL array, dimension (LDA, N) On entry, the N-by-N matrix A. On exit, A is overwritten by its real 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 (after sorting) for which SELECT is true. (Complex conjugate pairs for which SELECT is true for either eigenvalue count as 2.)

*WR*

WR is REAL array, dimension (N)

*WI*

WI is REAL array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

*VS*

VS is REAL array, dimension (LDVS,N) If JOBVS = 'V', VS contains the orthogonal 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 REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,3*N). Also, if SENSE = 'E' or 'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of selected eigenvalues computed by this routine. Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only returned if LWORK < max(1,3*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 bounds on the optimal sizes of the arrays WORK and IWORK, returns these values as the first entries of the WORK and IWORK arrays, and no error messages related to LWORK or LIWORK are issued by XERBLA.

*IWORK*

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

*LIWORK*

LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM). Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this may not be large enough. If LIWORK = -1, then a workspace query is assumed; the routine only calculates upper bounds on the optimal sizes of the arrays WORK and IWORK, returns these values as the first entries of the WORK and IWORK arrays, and no error messages related to LWORK or LIWORK are 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. > 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 WR and WI 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 sgeev (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

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

**Purpose: **

SGEEV computes for an N-by-N real 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 A 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 REAL 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).

*WR*

WR is REAL array, dimension (N)

*WI*

WI is REAL array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

*VL*

VL is REAL 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. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1).

*LDVL*

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

*VR*

VR is REAL 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. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and v(j+1) = VR(:,j) - i*VR(:,j+1).

*LDVR*

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

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,3*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*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.

*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 i+1:N of WR and WI contain eigenvalues which have converged.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

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

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

**Purpose: **

SGEEVX computes for an N-by-N real 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, i.e. 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 REAL 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 real Schur form of the balanced version of the input matrix A.

*LDA*

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

*WR*

WR is REAL array, dimension (N)

*WI*

WI is REAL array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

*VL*

VL is REAL 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. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1).

*LDVL*

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

*VR*

VR is REAL 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. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and v(j+1) = VR(:,j) - i*VR(:,j+1).

*LDVR*

LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1, and 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*

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 SENSE = 'N' or 'E', LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). 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.

*IWORK*

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

*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 WR and WI contain eigenvalues which have converged.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## 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 "standardized" 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*

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## 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)**

**Purpose: **

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 "standardized" 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*

*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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

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

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

**Purpose: **

SGGESX computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real 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)**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; 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 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 "standardized" 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 a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(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.

*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 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.

*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 numbers 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( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else LWORK >= max( 8*N, 6*N+16 ). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < max( 8*N, 6*N+16), 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.

*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 IWORK. If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise LIWORK >= N+6. 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is EPS * norm((A, B)) / RCONDE( 1 ). An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is EPS * norm((A, B)) / RCONDV( 2 ). See LAPACK User's Guide, section 4.11 for more information.

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

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

**Purpose: **

SGGEV computes for a pair of N-by-N real 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 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:**

*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 REAL 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 REAL 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).

*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. 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 alpha/beta. 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).

*VL*

VL is REAL 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 the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). 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 REAL 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 the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). 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. LWORK >= max(1,8*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.

*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 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: error return from STGEVC.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine sggev3 (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)**

**Purpose: **

SGGEV3 computes for a pair of N-by-N real 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 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:**

*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 REAL 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 REAL 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).

*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. 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 alpha/beta. 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).

*VL*

VL is REAL 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 the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). 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 REAL 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 the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). 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 If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, 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: error return from STGEVC.

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## subroutine sggevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer LDVL, real, 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, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)¶

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

**Purpose: **

SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. Optionally also, it 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 REAL 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 real 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 REAL 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 real 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).

*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. 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 ALPHA/BETA. 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).

*VL*

VL is REAL 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 the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector will be scaled so the largest component 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 REAL 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 the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each eigenvector will be scaled so the largest component 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. For a complex conjugate pair of eigenvalues two consecutive elements of RCONDE are set to the same value. Thus RCONDE(j), RCONDV(j), and the j-th columns of VL and VR all correspond to the j-th eigenpair. 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. For a complex eigenvector two consecutive elements of RCONDV are set to the same value. 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 BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', LWORK >= max(1,6*N). If SENSE = 'E', LWORK >= max(1,10*N). If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (N+6) 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 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: error return from STGEVC.

**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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