hbgv(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 NAME 💖

hbgv — {hb,sb}gv: eig, QR iteration.

okay bestie, welcome to the hbgv(3) page from the Library Functions Manual. This is the group chat for the {hb,sb}gv routines, and we are keeping every detail. no cap. ✨

✨ SYNOPSIS ✨

🌟 Functions 🌟

   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 💅

okay so the actual function tea lives in the docs below. No extra lore here, just the signatures and the parameter breakdowns. 🫶

🎀 Function Documentation 🎀

💖 subroutine chbgv — CHBGV 💖

   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

okay so 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. That's the whole assignment, bestie. 💅

🧾 Parameters

  • JOBZ — JOBZ is CHARACTER*1. = 'N': compute eigenvalues only; = 'V': compute eigenvalues and eigenvectors. Pick your fighter. ✨
  • UPLO — UPLO is CHARACTER*1. = 'U': upper triangles of A and B are stored; = 'L': lower triangles of A and B are stored. no cap.
  • 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. not me forgetting to warn you, bestie. 😭
  • 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. iconic.
  • 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 — DSBGV 💖

   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

okay so 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. periodt. 💜

🧾 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. it's giving "say bye to your data." 💀
  • 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 — SSBGV 💖

   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

okay so 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 ✨ (ssbgv bestie edition)

  • JOBZ — type CHARACTER*1. The vibe selector, fr:
    • 'N' → compute eigenvalues only;
    • 'V' → compute eigenvalues and eigenvectors.
  • UPLO — type CHARACTER*1. Tells us which triangle is living in the array:
    • 'U' → upper triangles of A and B are stored;
    • 'L' → lower triangles of A and B are stored.
  • N — type INTEGER. The order of matrices A and B. Constraint: N >= 0.
  • KA — type INTEGER. The number of superdiagonals of matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KA >= 0.
  • KB — type INTEGER. The number of superdiagonals of matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KB >= 0.
  • AB — type REAL array, dimension (LDAB, N). On entry, this holds 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 like this:
    • 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. Not me forgetting to warn you — that’s literally the manpage energy. 💀

  • LDAB — type INTEGER. The leading dimension of the array AB. Constraint: LDAB >= KA+1.
  • BB — type REAL array, dimension (LDBB, N). On entry, this holds 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 like this:
    • 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, this holds the factor S from the split Cholesky factorization B = S**T*S, as returned by SPBSTF. Slay. ✨

  • LDBB — type INTEGER. The leading dimension of the array BB. Constraint: LDBB >= KB+1.
  • W — type REAL array, dimension (N). If INFO = 0, this holds the eigenvalues in ascending order. Cute and sorted, we love to see it.
  • Z — type 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. Respect the boundaries, bestie. 🫶
  • LDZ — type INTEGER. The leading dimension of the array Z. Constraint: LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
  • WORK — type REAL array, dimension (3*N). The work array. It’s giving backstage crew — not the main character, but nothing happens without it.
  • INFO — type INTEGER. The receipts:
    • = 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.

💜 zhbgv subroutine 💜

   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. It’s giving complex eigenvalues with main character energy — but the math is doing the actual slaying. 💅

✨ Parameters ✨ (zhbgv edition)

  • JOBZ — type CHARACTER*1. Same selector energy:
    • 'N' → compute eigenvalues only;
    • 'V' → compute eigenvalues and eigenvectors.
  • UPLO — type CHARACTER*1. Which triangle are we storing?
    • 'U' → upper triangles of A and B are stored;
    • 'L' → lower triangles of A and B are stored.
  • N — type INTEGER. The order of matrices A and B. Constraint: N >= 0.
  • KA — type INTEGER. The number of superdiagonals of matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KA >= 0.
  • KB — type INTEGER. The number of superdiagonals of matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KB >= 0.
  • AB — type COMPLEX*16 array, dimension (LDAB, N). On entry, this holds 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 like this:
    • 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. Yes, bestie, it gets cooked. 🔥

  • LDAB — type INTEGER. The leading dimension of the array AB. Constraint: LDAB >= KA+1.
  • BB — type COMPLEX*16 array, dimension (LDBB, N). On entry, this holds 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 like this:
    • 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, this holds the factor S from the split Cholesky factorization B = S**H*S, as returned by ZPBSTF. Iconic factorization behavior. 🌟

  • LDBB — type INTEGER. The leading dimension of the array BB. Constraint: LDBB >= KB+1.
  • W — type DOUBLE PRECISION array, dimension (N). If INFO = 0, this holds the eigenvalues in ascending order.
  • Z — type 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. We stan a normalized eigenvector. 🎀
  • LDZ — type INTEGER. The leading dimension of the array Z. Constraint: LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
  • WORK — type COMPLEX*16 array, dimension (N). Work array. Essential, even if it doesn’t get a shoutout in the group chat.
  • RWORK — type DOUBLE PRECISION array, dimension (3*N). Real work array. Different lane, same mission.
  • INFO — type INTEGER. The final receipts:
    • = 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 🧚

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💖 That’s the footer tea, bestie: LAPACK, Version 3.12.0, hbgv(3) — periodt. 🫶✨

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