hpgvd(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 NAME 💖

hpgvd — {hp,sp}gvd: eig, divide and conquer. Straight from the Library Functions Manual, bestie. 🎀

✨ SYNOPSIS ✨

💅 Functions 💅

  • CHPGVD — the complex single-precision one:
subroutine chpgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork,
    rwork, lrwork, iwork, liwork, info)
  • DSPGVD — the double-precision real symmetric one:
subroutine dspgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork,
    iwork, liwork, info)
  • SSPGVD — the single-precision real symmetric one:
subroutine sspgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork,
    iwork, liwork, info)
  • ZHPGVD — the complex double-precision one:
subroutine zhpgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork,
    rwork, lrwork, iwork, liwork, info)

🌸 Detailed Description 🌸

The full details are living in the function docs below, bestie. That's where the actual lore is. 💖

🦋 Function Documentation 🦋

🎀 subroutine chpgvd 🎀

subroutine chpgvd (integer itype, character jobz, character uplo, integer n,
    complex, dimension( * ) ap, complex, dimension( * ) bp, real, dimension(
    * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension(
    * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork,
    integer, dimension( * ) iwork, integer liwork, integer info)

CHPGVD

💖 Purpose 💖

Okay so CHPGVD computes all the eigenvalues and, if you want them, the eigenvectors of a complex generalized Hermitian-definite eigenproblem. The form can be one of these:

  • A*x=(lambda)*B*x
  • A*Bx=(lambda)*x
  • B*A*x=(lambda)*x

Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. It's giving efficient, honestly. 💅

🎀 Parameters 🎀

  • ITYPE — INTEGER. Specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — CHARACTER*1:
    • 'N': compute eigenvalues only;
    • 'V': compute eigenvalues and eigenvectors.
  • UPLO — CHARACTER*1:
    • 'U': upper triangles of A and B are stored;
    • 'L': lower triangles of A and B are stored.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • AP — COMPLEX array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. 😭

  • BP — COMPLEX array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:
    • if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    • if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.

  • W — REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z — COMPLEX array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
    • if ITYPE = 1 or 2, Z**H*B*Z = I;
    • if ITYPE = 3, Z**H*inv(B)*Z = I.

If JOBZ = 'N', then Z is not referenced.

  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the required LWORK.
  • LWORK — INTEGER. The dimension of array WORK.
    • If N <= 1, LWORK >= 1.
    • If JOBZ = 'N' and N > 1, LWORK >= N.
    • If JOBZ = 'V' and N > 1, LWORK >= 2*N.
    • If LWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
  • RWORK — REAL array, dimension (MAX(1,LRWORK)). On exit, if INFO = 0, RWORK(1) returns the required LRWORK.
  • LRWORK — INTEGER. The dimension of array RWORK.
    • If N <= 1, LRWORK >= 1.
    • If JOBZ = 'N' and N > 1, LRWORK >= N.
    • If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
    • If LRWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
  • LIWORK — INTEGER. The dimension of array IWORK.
    • If JOBZ = 'N' or N <= 1, LIWORK >= 1.
    • If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
    • If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
  • INFO — INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: CPPTRF or CHPEVD returned an error code:
      • <= N: if INFO = i, CHPEVD 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 the leading principal minor of order i of B is not positive. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Not the vibe. 💀

👑 Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🎀 Contributors: 🎀

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 223 of file chpgvd.f.

🍓 subroutine dspgvd 🍓

subroutine dspgvd (integer itype, character jobz, character uplo, integer n,
    double precision, dimension( * ) ap, double precision, dimension( * )
    bp, double precision, dimension( * ) w, double precision, dimension(
    ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSPGVD

💖 Purpose 💖

DSPGVD computes all the eigenvalues, and optionally the eigenvectors, of a real generalized symmetric-definite eigenproblem. The form can be one of these:

  • A*x=(lambda)*B*x
  • A*Bx=(lambda)*x
  • B*A*x=(lambda)*x

Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. Same energy, real symmetric flavor. ✨

🎀 Parameters 🎀

  • ITYPE — INTEGER. Specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — CHARACTER*1:
    • 'N': compute eigenvalues only;
    • 'V': compute eigenvalues and eigenvectors.
  • UPLO — CHARACTER*1:
    • 'U': upper triangles of A and B are stored;
    • 'L': lower triangles of A and B are stored.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • AP — DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. 😭

  • BP — DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:
    • if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    • if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.

  • W — DOUBLE PRECISION array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
  • Z
  • Z — a DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V' and INFO = 0, this bad boy holds the eigenvector matrix Z. And bestie, the normalization is specific: if ITYPE = 1 or 2, then Z**T*B*Z = I; if ITYPE = 3, then Z**T*inv(B)*Z = I. If JOBZ = 'N', then Z is literally not referenced — it's just vibing, unbothered. 💅
  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N).
  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the required LWORK. It's giving "I did the math so you don't have to" energy. ✨
  • LWORK — INTEGER, the dimension of the array WORK. The rules, respectfully:
    • If N <= 1, LWORK >= 1.
    • If JOBZ = 'N' and N > 1, LWORK >= 2*N.
    • If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

And if LWORK = -1, then a workspace query is assumed — the routine only calculates the required sizes of the WORK and IWORK arrays, 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. So like, a little sneak peek instead of a full computation. 🫶

  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
  • LIWORK — INTEGER, the dimension of the array IWORK. The tea:
    • If JOBZ = 'N' or N <= 1, LIWORK >= 1.
    • If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK and IWORK arrays, 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.

  • INFO — INTEGER. The receipts:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: DPPTRF or DSPEVD returned an error code:
      • <= N: if INFO = i, DSPEVD 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 the leading principal minor of order i of B is not positive. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Rough. 💀

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 202 of file dspgvd.f.

🌸 subroutine sspgvd 🌸

subroutine sspgvd (integer itype, character jobz, character uplo, integer n,
    real, dimension( * ) ap, real, dimension( * ) bp, real, dimension( * )
    w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work,
    integer lwork, integer, dimension( * ) iwork, integer liwork, integer
    info)

SSPGVD — ok so this is the single-precision bestie of the one above, same whole entire situation, different letter. 💖

Purpose:

SSPGVD computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. Iconic behavior, no notes. ✨

Parameters

  • ITYPE — INTEGER. Specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — CHARACTER*1:
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • UPLO — CHARACTER*1:
    • = 'U': Upper triangles of A and B are stored;
    • = 'L': Lower triangles of A and B are stored.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • AP — REAL array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. Gone. Poof. 🦋

  • BP — REAL array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:
    • if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
    • if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B. She transforms but she stays in the same outfit. 💅

  • W — REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order. Sorted, as one should be. 👑
  • Z — REAL array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If JOBZ = 'N', then Z is not referenced.
  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the required LWORK.
  • LWORK — INTEGER, the dimension of the array WORK.
    • If N <= 1, LWORK >= 1.
    • If JOBZ = 'N' and N > 1, LWORK >= 2*N.
    • If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK and IWORK arrays, 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 — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
  • LIWORK — INTEGER, the dimension of the array IWORK.
    • If JOBZ = 'N' or N <= 1, LIWORK >= 1.
    • If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK and IWORK arrays, 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.

  • INFO — INTEGER.
    • = 0: successful exit 🎀
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: SPPTRF or SSPEVD returned an error code:
      • <= N: if INFO = i, SSPEVD 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 the leading principal minor of order i of B is not positive. 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.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 202 of file sspgvd.f.

🦋 subroutine zhpgvd 🦋

subroutine zhpgvd (integer itype, character jobz, character uplo, integer n,
    complex*16, dimension( * ) ap, complex*16, dimension( * ) bp, double
    precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer
    ldz, complex*16, dimension( * ) work, integer lwork, double precision,
    dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork,
    integer liwork, integer info)

ZHPGVD — now we're in complex double precision territory, and she brought extra workspace arrays with her because she's high maintenance in the most valid way possible. 💜

Purpose:

ZHPGVD computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm.

Note the glow-up: this one says Hermitian, not symmetric — because complex numbers said "we do things differently here." 🌈

Parameters

  • ITYPE — INTEGER. Specifies the problem type to be solved:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — CHARACTER*1:
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • UPLO — CHARACTER*1:
    • = 'U': Upper triangles of A and B are stored;
    • = 'L': Lower triangles of A and B are stored.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • AP

💅 AP

AP is a COMPLEX*16 array, dimension (N*(N+1)/2). On entry, it holds the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP like this:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, the contents of AP are destroyed 😭 so don't expect your data to still be there, bestie!

🦋 BP

BP is a COMPLEX*16 array, dimension (N*(N+1)/2). On entry, it holds the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP like this:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, it contains the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B. ✨

🌸 W

W is a DOUBLE PRECISION array, dimension (N). If INFO = 0, it holds the eigenvalues in ascending order. 💖

🎀 Z

Z is a COMPLEX*16 array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

  • if ITYPE = 1 or 2, Z**H*B*Z = I;
  • if ITYPE = 3, Z**H*inv(B)*Z = I.

If JOBZ = 'N', then Z is not referenced at all. 🙈

🧚 LDZ

LDZ is INTEGER. It's the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). Slay that dimension check! 👑

🔥 WORK

WORK is a COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the required LWORK. 💅

⭐ LWORK

LWORK is INTEGER. It's the dimension of the array WORK.

  • If N <= 1, LWORK >= 1.
  • If JOBZ = 'N' and N > 1, LWORK >= N.
  • If JOBZ = 'V' and N > 1, LWORK >= 2*N.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. So don't freak out, it's just a query! 😌

💜 RWORK

RWORK is a DOUBLE PRECISION array, dimension (MAX(1,LRWORK)). On exit, if INFO = 0, RWORK(1) returns the required LRWORK. 🫶

🌈 LRWORK

LRWORK is INTEGER. It's the dimension of array RWORK.

  • If N <= 1, LRWORK >= 1.
  • If JOBZ = 'N' and N > 1, LRWORK >= N.
  • If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.

If LRWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. Same vibe as LWORK, bestie! ✨

🍓 IWORK

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

🐚 LIWORK

LIWORK is INTEGER. It's the dimension of array IWORK.

  • If JOBZ = 'N' or N <= 1, LIWORK >= 1.
  • If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 🎀

💀 INFO

INFO is INTEGER.

  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 😬
  • > 0: ZPPTRF or ZHPEVD returned an error code:
    • <= N: if INFO = i, ZHPEVD failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not convergeto zero; 💀
    • > N: if INFO = N + i, for 1 <= i <= n, then the leading principal minor of order i of B is not positive. 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.

🫶 Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

📌 Definition at line 223 of file zhpgvd.f.

✨ Author

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And that's a wrap, bestie! This iconic page is from LAPACK Version 3.12.0 and it's the hpgvd(3) man page. Stay slay, stay hermitian, and may your eigenvalues always converge! 💖✨👑

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