--- name: "hbgv" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/hbgv" --- # hbgv(3) {hb,sb}gv: eig, QR iteration ## NAME hbgv - {hb,sb}gv: eig, QR iteration ## SYNOPSIS ### Functions subroutine **chbgv** (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)\ **CHBGV** subroutine **dsbgv** (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info)\ **DSBGV** subroutine **ssbgv** (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info)\ **SSBGV** subroutine **zhbgv** (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)\ **ZHBGV** ## Detailed Description ## Function Documentation ### subroutine chbgv (character jobz, character uplo, integer n, integer ka, integer kb, complex, dimension( ldab, \* ) ab, integer ldab, complex, dimension( ldbb, \* ) bb, integer ldbb, real, dimension( \* ) w, complex, dimension( ldz, \* ) z, integer ldz, complex, dimension( \* ) work, real, dimension( \* ) rwork, integer info) **CHBGV** **Purpose:** ``` !> !> CHBGV computes all the eigenvalues, and optionally, the eigenvectors !> of a complex generalized Hermitian-definite banded eigenproblem, of !> the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian !> and banded, and B is also positive definite. !> ``` **Parameters** *JOBZ* ``` !> JOBZ is CHARACTER*1 !> = 'N': Compute eigenvalues only; !> = 'V': Compute eigenvalues and eigenvectors. !> ``` *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangles of A and B are stored; !> = 'L': Lower triangles of A and B are 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'. 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 contents of AB are destroyed. !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KA+1. !> ``` *BB* ``` !> BB is COMPLEX array, dimension (LDBB, N) !> On entry, the upper or lower triangle of the Hermitian band !> matrix B, stored in the first kb+1 rows of the array. The !> j-th column of B is stored in the j-th column of the array BB !> as follows: !> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; !> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). !> !> On exit, the factor S from the split Cholesky factorization !> B = S**H*S, as returned by CPBSTF. !> ``` *LDBB* ``` !> LDBB is INTEGER !> The leading dimension of the array BB. LDBB >= KB+1. !> ``` *W* ``` !> W is REAL array, dimension (N) !> If INFO = 0, the eigenvalues in ascending order. !> ``` *Z* ``` !> Z is COMPLEX array, dimension (LDZ, N) !> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of !> eigenvectors, with the i-th column of Z holding the !> eigenvector associated with W(i). The eigenvectors are !> normalized so that Z**H*B*Z = I. !> If JOBZ = 'N', then Z is not referenced. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= N. !> ``` *WORK* ``` !> WORK is COMPLEX array, dimension (N) !> ``` *RWORK* ``` !> RWORK is REAL array, dimension (3*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 algorithm failed to converge: !> i off-diagonal elements of an intermediate !> tridiagonal form did not converge to zero; !> > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF !> returned INFO = i: B is not positive definite. !> The factorization of B could not be completed and !> no eigenvalues or eigenvectors were computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **181** of file **chbgv.f**. ### subroutine dsbgv (character jobz, 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( \* ) w, double precision, dimension( ldz, \* ) z, integer ldz, double precision, dimension( \* ) work, integer info) **DSBGV** **Purpose:** ``` !> !> DSBGV computes all the eigenvalues, and optionally, the eigenvectors !> of a real generalized symmetric-definite banded eigenproblem, of !> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric !> and banded, and B is also positive definite. !> ``` **Parameters** *JOBZ* ``` !> JOBZ is CHARACTER*1 !> = 'N': Compute eigenvalues only; !> = 'V': Compute eigenvalues and eigenvectors. !> ``` *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangles of A and B are stored; !> = 'L': Lower triangles of A and B are 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'. 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 contents of AB are destroyed. !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KA+1. !> ``` *BB* ``` !> BB is DOUBLE PRECISION array, dimension (LDBB, N) !> On entry, the upper or lower triangle of the symmetric band !> matrix B, stored in the first kb+1 rows of the array. The !> j-th column of B is stored in the j-th column of the array BB !> as follows: !> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; !> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). !> !> On exit, the factor S from the split Cholesky factorization !> B = S**T*S, as returned by DPBSTF. !> ``` *LDBB* ``` !> LDBB is INTEGER !> The leading dimension of the array BB. LDBB >= KB+1. !> ``` *W* ``` !> W is DOUBLE PRECISION array, dimension (N) !> If INFO = 0, the eigenvalues in ascending order. !> ``` *Z* ``` !> Z is DOUBLE PRECISION array, dimension (LDZ, N) !> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of !> eigenvectors, with the i-th column of Z holding the !> eigenvector associated with W(i). The eigenvectors are !> normalized so that Z**T*B*Z = I. !> If JOBZ = 'N', then Z is not referenced. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= N. !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (3*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 algorithm failed to converge: !> i off-diagonal elements of an intermediate !> tridiagonal form did not converge to zero; !> > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF !> returned INFO = i: B is not positive definite. !> The factorization of B could not be completed and !> no eigenvalues or eigenvectors were computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **175** of file **dsbgv.f**. ### subroutine ssbgv (character jobz, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, \* ) ab, integer ldab, real, dimension( ldbb, \* ) bb, integer ldbb, real, dimension( \* ) w, real, dimension( ldz, \* ) z, integer ldz, real, dimension( \* ) work, integer info) **SSBGV** **Purpose:** ``` !> !> SSBGV computes all the eigenvalues, and optionally, the eigenvectors !> of a real generalized symmetric-definite banded eigenproblem, of !> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric !> and banded, and B is also positive definite. !> ``` **Parameters** *JOBZ* ``` !> JOBZ is CHARACTER*1 !> = 'N': Compute eigenvalues only; !> = 'V': Compute eigenvalues and eigenvectors. !> ``` *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangles of A and B are stored; !> = 'L': Lower triangles of A and B are 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'. 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 contents of AB are destroyed. !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KA+1. !> ``` *BB* ``` !> BB is REAL array, dimension (LDBB, N) !> On entry, the upper or lower triangle of the symmetric band !> matrix B, stored in the first kb+1 rows of the array. The !> j-th column of B is stored in the j-th column of the array BB !> as follows: !> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; !> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). !> !> On exit, the factor S from the split Cholesky factorization !> B = S**T*S, as returned by SPBSTF. !> ``` *LDBB* ``` !> LDBB is INTEGER !> The leading dimension of the array BB. LDBB >= KB+1. !> ``` *W* ``` !> W is REAL array, dimension (N) !> If INFO = 0, the eigenvalues in ascending order. !> ``` *Z* ``` !> Z is REAL array, dimension (LDZ, N) !> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of !> eigenvectors, with the i-th column of Z holding the !> eigenvector associated with W(i). The eigenvectors are !> normalized so that Z**T*B*Z = I. !> If JOBZ = 'N', then Z is not referenced. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= N. !> ``` *WORK* ``` !> WORK is REAL array, dimension (3*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 algorithm failed to converge: !> i off-diagonal elements of an intermediate !> tridiagonal form did not converge to zero; !> > N: if INFO = N + i, for 1 <= i <= N, then SPBSTF !> returned INFO = i: B is not positive definite. !> The factorization of B could not be completed and !> no eigenvalues or eigenvectors were computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **175** of file **ssbgv.f**. ### subroutine zhbgv (character jobz, character uplo, integer n, integer ka, integer kb, complex\*16, dimension( ldab, \* ) ab, integer ldab, complex\*16, dimension( ldbb, \* ) bb, integer ldbb, double precision, dimension( \* ) w, complex\*16, dimension( ldz, \* ) z, integer ldz, complex\*16, dimension( \* ) work, double precision, dimension( \* ) rwork, integer info) **ZHBGV** **Purpose:** ``` !> !> ZHBGV computes all the eigenvalues, and optionally, the eigenvectors !> of a complex generalized Hermitian-definite banded eigenproblem, of !> the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian !> and banded, and B is also positive definite. !> ``` **Parameters** *JOBZ* ``` !> JOBZ is CHARACTER*1 !> = 'N': Compute eigenvalues only; !> = 'V': Compute eigenvalues and eigenvectors. !> ``` *UPLO* ``` !> UPLO is CHARACTER*1 !> = 'U': Upper triangles of A and B are stored; !> = 'L': Lower triangles of A and B are 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'. KB >= 0. !> ``` *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 contents of AB are destroyed. !> ``` *LDAB* ``` !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KA+1. !> ``` *BB* ``` !> BB is COMPLEX*16 array, dimension (LDBB, N) !> On entry, the upper or lower triangle of the Hermitian band !> matrix B, stored in the first kb+1 rows of the array. The !> j-th column of B is stored in the j-th column of the array BB !> as follows: !> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; !> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). !> !> On exit, the factor S from the split Cholesky factorization !> B = S**H*S, as returned by ZPBSTF. !> ``` *LDBB* ``` !> LDBB is INTEGER !> The leading dimension of the array BB. LDBB >= KB+1. !> ``` *W* ``` !> W is DOUBLE PRECISION array, dimension (N) !> If INFO = 0, the eigenvalues in ascending order. !> ``` *Z* ``` !> Z is COMPLEX*16 array, dimension (LDZ, N) !> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of !> eigenvectors, with the i-th column of Z holding the !> eigenvector associated with W(i). The eigenvectors are !> normalized so that Z**H*B*Z = I. !> If JOBZ = 'N', then Z is not referenced. !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= N. !> ``` *WORK* ``` !> WORK is COMPLEX*16 array, dimension (N) !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION array, dimension (3*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 algorithm failed to converge: !> i off-diagonal elements of an intermediate !> tridiagonal form did not converge to zero; !> > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF !> returned INFO = i: B is not positive definite. !> The factorization of B could not be completed and !> no eigenvalues or eigenvectors were computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **181** of file **zhbgv.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.