# hbgst - Man Page

{hb,sb}gst: reduction to standard form, banded

## Synopsis

### Functions

subroutine **chbgst** (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info)**CHBGST**

subroutine **dsbgst** (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)**DSBGST**

subroutine **ssbgst** (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)**SSBGST**

subroutine **zhbgst** (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info)**ZHBGST**

## Detailed Description

## Function Documentation

### subroutine chbgst (character vect, character uplo, integer n, integer ka, integer kb, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldbb, * ) bb, integer ldbb, complex, dimension( ldx, * ) x, integer ldx, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

**CHBGST**

**Purpose:**

CHBGST reduces a complex Hermitian-definite banded generalized eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, such that C has the same bandwidth as A. B must have been previously factorized as S**H*S by CPBSTF, using a split Cholesky factorization. A is overwritten by C = X**H*A*X, where X = S**(-1)*Q and Q is a unitary matrix chosen to preserve the bandwidth of A.

**Parameters***VECT*VECT is CHARACTER*1 = 'N': do not form the transformation matrix X; = 'V': form X.

*UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

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

*KA*KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

*KB*KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0.

*AB*AB is COMPLEX array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the transformed matrix X**H*A*X, stored in the same format as A.

*LDAB*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.

*BB*BB is COMPLEX array, dimension (LDBB,N) The banded factor S from the split Cholesky factorization of B, as returned by CPBSTF, stored in the first kb+1 rows of the array.

*LDBB*LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.

*X*X is COMPLEX array, dimension (LDX,N) If VECT = 'V', the n-by-n matrix X. If VECT = 'N', the array X is not referenced.

*LDX*LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

*WORK*WORK is COMPLEX array, dimension (N)

*RWORK*RWORK is REAL array, dimension (N)

*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.

Definition at line **163** of file **chbgst.f**.

### subroutine dsbgst (character vect, character uplo, integer n, integer ka, integer kb, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) work, integer info)

**DSBGST**

**Purpose:**

DSBGST reduces a real symmetric-definite banded generalized eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, such that C has the same bandwidth as A. B must have been previously factorized as S**T*S by DPBSTF, using a split Cholesky factorization. A is overwritten by C = X**T*A*X, where X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the bandwidth of A.

**Parameters***VECT*VECT is CHARACTER*1 = 'N': do not form the transformation matrix X; = 'V': form X.

*UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

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

*KA*KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

*KB*KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0.

*AB*AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the transformed matrix X**T*A*X, stored in the same format as A.

*LDAB*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.

*BB*BB is DOUBLE PRECISION array, dimension (LDBB,N) The banded factor S from the split Cholesky factorization of B, as returned by DPBSTF, stored in the first KB+1 rows of the array.

*LDBB*LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.

*X*X is DOUBLE PRECISION array, dimension (LDX,N) If VECT = 'V', the n-by-n matrix X. If VECT = 'N', the array X is not referenced.

*LDX*LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

*WORK*WORK is DOUBLE PRECISION array, dimension (2*N)

*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.

Definition at line **157** of file **dsbgst.f**.

### subroutine ssbgst (character vect, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) work, integer info)

**SSBGST**

**Purpose:**

SSBGST reduces a real symmetric-definite banded generalized eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, such that C has the same bandwidth as A. B must have been previously factorized as S**T*S by SPBSTF, using a split Cholesky factorization. A is overwritten by C = X**T*A*X, where X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the bandwidth of A.

**Parameters***VECT*VECT is CHARACTER*1 = 'N': do not form the transformation matrix X; = 'V': form X.

*UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

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

*KA*KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

*KB*KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0.

*AB*AB is REAL array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the transformed matrix X**T*A*X, stored in the same format as A.

*LDAB*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.

*BB*BB is REAL array, dimension (LDBB,N) The banded factor S from the split Cholesky factorization of B, as returned by SPBSTF, stored in the first KB+1 rows of the array.

*LDBB*LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.

*X*X is REAL array, dimension (LDX,N) If VECT = 'V', the n-by-n matrix X. If VECT = 'N', the array X is not referenced.

*LDX*LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

*WORK*WORK 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.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **157** of file **ssbgst.f**.

### subroutine zhbgst (character vect, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16, dimension( ldx, * ) x, integer ldx, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

**ZHBGST**

**Purpose:**

ZHBGST reduces a complex Hermitian-definite banded generalized eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, such that C has the same bandwidth as A. B must have been previously factorized as S**H*S by ZPBSTF, using a split Cholesky factorization. A is overwritten by C = X**H*A*X, where X = S**(-1)*Q and Q is a unitary matrix chosen to preserve the bandwidth of A.

**Parameters***VECT*VECT is CHARACTER*1 = 'N': do not form the transformation matrix X; = 'V': form X.

*UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

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

*KA**KB**AB*AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the transformed matrix X**H*A*X, stored in the same format as A.

*LDAB*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.

*BB*BB is COMPLEX*16 array, dimension (LDBB,N) The banded factor S from the split Cholesky factorization of B, as returned by ZPBSTF, stored in the first kb+1 rows of the array.

*LDBB*LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.

*X*X is COMPLEX*16 array, dimension (LDX,N) If VECT = 'V', the n-by-n matrix X. If VECT = 'N', the array X is not referenced.

*LDX**WORK*WORK is COMPLEX*16 array, dimension (N)

*RWORK*RWORK is DOUBLE PRECISION array, dimension (N)

*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.

Definition at line **163** of file **zhbgst.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.