# gghrd - Man Page

gghrd: reduction to Hessenberg

## Synopsis

### Functions

subroutine **cgghrd** (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)**CGGHRD**

subroutine **dgghrd** (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)**DGGHRD**

subroutine **sgghrd** (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)**SGGHRD**

subroutine **zgghrd** (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)**ZGGHRD**

## Detailed Description

## Function Documentation

### subroutine cgghrd (character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer info)

**CGGHRD**

**Purpose:**

CGGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = lambda*B*x, and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**H*A*Z = H and transforms B to another upper triangular matrix T: Q**H*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**H*x. The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H If Q1 is the unitary matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then CGGHRD reduces the original problem to generalized Hessenberg form.

**Parameters***COMPQ*COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.

*COMPZ*COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned; = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned.

*N*N is INTEGER The order of the matrices A and B. N >= 0.

*ILO*ILO is INTEGER

*IHI*IHI is INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*A*A is COMPLEX array, dimension (LDA, N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.

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

*B*B is COMPLEX array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero.

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

*Q*Q is COMPLEX array, dimension (LDQ, N) On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

*Z*Z is COMPLEX array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and van Loan (Johns Hopkins Press).

Definition at line **202** of file **cgghrd.f**.

### subroutine dgghrd (character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, integer info)

**DGGHRD**

**Purpose:**

DGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = lambda*B*x, and B is typically made upper triangular by computing its QR factorization and moving the orthogonal matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**T*A*Z = H and transforms B to another upper triangular matrix T: Q**T*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**T*x. The orthogonal matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then DGGHRD reduces the original problem to generalized Hessenberg form.

**Parameters***COMPQ*COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = 'V': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned.

*COMPZ*COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.

*N*N is INTEGER The order of the matrices A and B. N >= 0.

*ILO*ILO is INTEGER

*IHI*IHI is INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*A*A is DOUBLE PRECISION array, dimension (LDA, N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.

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

*B*B is DOUBLE PRECISION array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**T B Z. The elements below the diagonal are set to zero.

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

*Q*Q is DOUBLE PRECISION array, dimension (LDQ, N) On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the orthogonal matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

*Z*Z is DOUBLE PRECISION array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I', the orthogonal matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and Van Loan (Johns Hopkins Press.)

Definition at line **205** of file **dgghrd.f**.

### subroutine sgghrd (character compq, character compz, integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, integer info)

**SGGHRD**

**Purpose:**

SGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = lambda*B*x, and B is typically made upper triangular by computing its QR factorization and moving the orthogonal matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**T*A*Z = H and transforms B to another upper triangular matrix T: Q**T*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**T*x. The orthogonal matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then SGGHRD reduces the original problem to generalized Hessenberg form.

**Parameters***COMPQ*COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = 'V': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned.

*COMPZ*COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.

*N*N is INTEGER The order of the matrices A and B. N >= 0.

*ILO*ILO is INTEGER

*IHI*IHI is INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to SGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*A*A is REAL array, dimension (LDA, N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.

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

*B*B is REAL array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**T B Z. The elements below the diagonal are set to zero.

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

*Q*Q is REAL array, dimension (LDQ, N) On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the orthogonal matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

*Z*Z is REAL array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I', the orthogonal matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and Van Loan (Johns Hopkins Press.)

Definition at line **205** of file **sgghrd.f**.

### subroutine zgghrd (character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer info)

**ZGGHRD**

**Purpose:**

ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = lambda*B*x, and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**H*A*Z = H and transforms B to another upper triangular matrix T: Q**H*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**H*x. The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H If Q1 is the unitary matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then ZGGHRD reduces the original problem to generalized Hessenberg form.

**Parameters***COMPQ*COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.

*COMPZ*COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned; = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned.

*N*N is INTEGER The order of the matrices A and B. N >= 0.

*ILO*ILO is INTEGER

*IHI*IHI is INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

*A*A is COMPLEX*16 array, dimension (LDA, N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.

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

*B*B is COMPLEX*16 array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero.

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

*Q*Q is COMPLEX*16 array, dimension (LDQ, N) On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

*LDQ**Z*Z is COMPLEX*16 array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

*LDZ**INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and van Loan (Johns Hopkins Press).

Definition at line **202** of file **zgghrd.f**.

## Author

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