hpgvx(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 hpgvx(3) 💖

okay so this is a page from the Library Functions Manual — the icon herself, hpgvx(3). no cap, this is the {hp,sp}gvx family: eig, bisection. that's the whole name section, bestie, we're already done with it. slay.

✨ SYNOPSIS ✨

🎀 Functions

  • subroutine chpgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) CHPGVX
  • subroutine dspgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) DSPGVX
  • subroutine sspgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) SSPGVX
  • subroutine zhpgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) ZHPGVX

🦋 Detailed Description

literally the whole "detailed description" is just the function docs below. no extra lore, bestie. keep scrolling 💅

👑 Function Documentation

🌸 subroutine chpgvx

subroutine chpgvx (integer itype, character jobz, character range, character
    uplo, integer n, complex, dimension( * ) ap, complex, dimension( * ) bp,
    real vl, real vu, integer il, integer iu, real abstol, integer m, real,
    dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex,
    dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * )
    iwork, integer, dimension( * ) ifail, integer info)

CHPGVX 🩷

⭐ Purpose

okay so CHPGVX computes selected eigenvalues and, if you want them, eigenvectors of a complex generalized Hermitian-definite eigenproblem. the forms are:

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. you can pick which eigenvalues/eigenvectors you get by specifying either a range of values or a range of indices for the ones you want. choose your own adventure, but make it linear algebra ✨

🌈 Parameters

  • ITYPE — INTEGER. specifies the problem type to be solved, like:
    • = 1: A*x = (lambda)*B*x
    • = 2: A*B*x = (lambda)*x
    • = 3: B*A*x = (lambda)*x
  • JOBZ — CHARACTER*1. do you want the eigenvectors or not, that is THE question:
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • RANGE — CHARACTER*1. which ones are we finding:
    • = 'A': all eigenvalues will be found;
    • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found;
    • = 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO — CHARACTER*1. which triangle did you actually store:
    • = '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. 💀 she's gone, bestie. don't come looking for her.
  • 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. glow up but make it factorization 🫶

  • VL — REAL. if RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. not referenced if RANGE = 'A' or 'I'.
  • VU — REAL. if RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. not referenced if RANGE = 'A' or 'I'.
  • IL — INTEGER. if RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. not referenced if RANGE = 'A' or 'V'.
  • IU — INTEGER. if RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — REAL. the absolute error tolerance for the eigenvalues. an approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. if ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form.

eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. so like, don't set it to zero and then act surprised, okay?

if this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). that's the glow-up fix 💅
  • M — INTEGER. the total number of eigenvalues found. 0 <= M <= N. if RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — REAL array, dimension (N). on normal exit, the first M elements contain the selected eigenvalues in ascending order. she's organized, she's sorted, she's thriving ✨
  • Z — COMPLEX array, dimension (LDZ, N). if JOBZ = 'N', then Z is not referenced. if JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL.

Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. fr fr, size it generously. no cap.
  • LDZ — INTEGER. the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — COMPLEX array, dimension (2*N).
  • RWORK — REAL array, dimension (7*N).
  • IWORK — INTEGER array, dimension (5*N).
  • IFAIL — INTEGER array, dimension (N). if JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. if INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. if JOBZ = 'N', then IFAIL is not referenced.
  • INFO — INTEGER. the whole personality, honestly:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: CPPTRF or CHPEVX returned an error code:
      • <= N: if INFO = i, CHPEVX failed to converge; — i eigenvectors failed to converge. their indices are stored in array IFAIL.
      • > 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 day for everyone.

🧚 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 274 of file chpgvx.f. 🔥

🌸 subroutine dspgvx

subroutine dspgvx (integer itype, character jobz, character range, character
    uplo, integer n, double precision, dimension( * ) ap, double precision,
    dimension( * ) bp, double precision vl, double precision vu, integer il,
    integer iu, double precision abstol, integer m, double precision,
    dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz,
    double precision, dimension( * ) work, integer, dimension( * ) iwork,
    integer, dimension( * ) ifail, integer info)

DSPGVX 🩷

⭐ Purpose

DSPGVX computes selected eigenvalues, and optionally, 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 storage, and B is also positive definite. eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. same energy, real numbers, still iconic 💜

🌈 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.
  • RANGE
  • RANGE — okay bestie, this one is a CHARACTER*1 and it chooses the eigenvalue hunt mode 💅
    • 'A': all eigenvalues will be found.
    • 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO — another CHARACTER*1, and it tells us which triangle is living in the arrays:
    • 'U': Upper triangle of A and B are stored;
    • 'L': Lower triangle of A and B are stored.
  • N — INTEGER, the order of the matrix pencil (A,B). N >= 0. No cap 🫶
  • AP — DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, this is 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 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. poof, gone 💀

  • BP — DOUBLE PRECISION array, dimension (N*(N+1)/2). On entry, this is 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 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, BP holds 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. iconic behavior ✨

  • VL — DOUBLE PRECISION. If RANGE='V', this is the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — DOUBLE PRECISION. If RANGE='V', this is the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL — INTEGER. If RANGE='I', this is the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • IU — INTEGER. If RANGE='I', this is the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — DOUBLE PRECISION. This is the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). bestie, this is the accuracy dial, don't skip it 🎀

  • M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — DOUBLE PRECISION array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. literally the main characters 👑
  • Z — DOUBLE PRECISION array, dimension (LDZ, max(1,M)). If JOBZ = 'N', then Z is not referenced. If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. no cap, plan ahead 🧚

  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — DOUBLE PRECISION array, dimension (8*N).
  • IWORK — INTEGER array, dimension (5*N).
  • IFAIL — INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
  • INFO — INTEGER. Here's the tea ☕
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: DPPTRF or DSPEVX returned an error code:
      • <= N: if INFO = i, DSPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL.
      • > 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 269 of file dspgvx.f.

   subroutine sspgvx (integer itype, character jobz, character range, character
       uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, real
       vl, real vu, integer il, integer iu, real abstol, integer m, real,
       dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real,
       dimension( * ) work, integer, dimension( * ) iwork, integer, dimension(
       * ) ifail, integer info)
       SSPGVX

Purpose

SSPGVX computes selected eigenvalues, and optionally, 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 storage, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. we love a flexible icon 🌈

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.
  • RANGE — CHARACTER*1:
    • 'A': all eigenvalues will be found.
    • 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
    • 'I': the IL-th through IU-th eigenvalues will be found.
  • UPLO — CHARACTER*1:
    • 'U': Upper triangle of A and B are stored;
    • 'L': Lower triangle of A and B are stored.
  • N — INTEGER. The order of the matrix pencil (A,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. rip to AP 🙈

  • 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.

  • VL — REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — REAL. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL —
  • IL — INTEGER energy. Only relevant when RANGE='I', and then it's the index of the smallest eigenvalue you want back. Constraint, no cap: 1 <= IL <= IU <= N, if N > 0; and IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V' — so don't stress about it then, bestie 🫶
  • IU — INTEGER too. If RANGE='I', this is the index of the largest eigenvalue to be returned. Same rules, same energy: 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. And again — not referenced if RANGE = 'A' or 'V' ✨
  • ABSTOL — a REAL, and it's the absolute error tolerance for the eigenvalues. Here's the vibe check on convergence: an approximate eigenvalue gets accepted as converged when it's determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| is used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Okay so this is the part people always get wrong 😭 Eigenvalues get computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. And if this routine comes back with INFO>0, meaning some eigenvectors did not converge — try setting ABSTOL to 2*SLAMCH('S'). That's the move 💅
  • M — INTEGER. The total number of eigenvalues found, and 0 <= M <= N. If RANGE = 'A', then M = N; and if RANGE = 'I', then M = IU-IL+1. Easy math, slay 🎀
  • W — REAL array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues, in ascending order. Cute and sorted, we love to see it ⭐
  • Z — REAL array, dimension (LDZ, max(1,M)). If JOBZ = 'N', then Z is not referenced (it's just sitting there, minding its business 🙈). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).

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.

And if an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL.

Important, bestie: the user must ensure that at least max(1,M) columns are supplied in the array Z. If RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Plan ahead, don't get caught lacking 💀
  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N). Respect the dimension, it's giving main character energy 👑
  • WORK — REAL array, dimension (8*N). That's your scratch space, go off 🧚
  • IWORK — INTEGER array, dimension (5*N). Integer scratch space, equally iconic ✨
  • IFAIL — INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero (clean, no failures, we love a peaceful life 😌). If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge — so you know exactly which ones ghosted you 😭. If JOBZ = 'N', then IFAIL is not referenced.
  • INFO — INTEGER, the moment of truth:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value (you passed something cursed, ngl)
    • > 0: SPPTRF or SSPEVX returned an error code:
      • <= N: if INFO = i, SSPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL.
      • > 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 💀 So like… the whole thing is a wash, start over.

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 269 of file sspgvx.f.

💜 subroutine zhpgvx

subroutine zhpgvx (integer itype, character jobz, character range, character
    uplo, integer n, complex*16, dimension( * ) ap, complex*16, dimension( *
    ) bp, double precision vl, double precision vu, integer il, integer iu,
    double precision abstol, integer m, double precision, dimension( * ) w,
    complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( *
    ) work, double precision, dimension( * ) rwork, integer, dimension( * )
    iwork, integer, dimension( * ) ifail, integer info)

ZHPGVX 🌸

Purpose:

!>
!> ZHPGVX computes selected eigenvalues and, optionally, 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.  Eigenvalues and eigenvectors can be selected by
!> specifying either a range of values or a range of indices for the
!> desired eigenvalues.
!>

Okay so to translate: ZHPGVX computes selected eigenvalues and, optionally, 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. It's giving ✨custom order✨ and honestly? obsessed 💖

✨ 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. (the full package, periodt 💅)
  • RANGE — CHARACTER*1.
    • = 'A': all eigenvalues will be found;
    • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found;
    • = 'I': the IL-th through IU-th eigenvalues will be found.
  • 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*16 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 😭 So don't come crying to me if you needed them later — it says what it says.
  • BP — COMPLEX*16 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. So BP gets a glow-up, not a deletion — noted 🦋
  • VL — DOUBLE PRECISION. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Constraint: VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — DOUBLE PRECISION. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Constraint: VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL — INTEGER. If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • IU — INTEGER. If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
  • ABSTOL — DOUBLE PRECISION. The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form.

Same warning as before because it matters, bestie: eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). That's the fix, iykyk 🔥
  • M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — DOUBLE PRECISION array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. Sorted, stunning, no notes 💜
  • Z —

Z — okay so Z is a COMPLEX*16 array, dimension (LDZ, N). If JOBZ = 'N', then Z is literally not referenced — it's sitting this one out, no cap.

If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL.

Note, bestie: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. 🫶

LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

WORK — WORK is a COMPLEX*16 array, dimension (2*N).

RWORK — RWORK is a DOUBLE PRECISION array, dimension (7*N).

IWORK — IWORK is an INTEGER array, dimension (5*N).

IFAIL — IFAIL is an INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

INFO — INFO is INTEGER.

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: ZPPTRF or ZHPEVX returned an error code:
    • <= N: if INFO = i, ZHPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL.
    • > 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 274 of file zhpgvx.f.

🦋 AUTHOR 🦋

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