pteqr(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

okay so bestie, this is pteqr(3) from the Library Functions Manual — and she’s giving major linear algebra main character energy. 💅✨

💅 NAME 💅

pteqr - pteqr: eig, positive definite tridiagonal

that’s the whole name, no cap. 🎀

✨ SYNOPSIS ✨

🫶 Functions 🫶

subroutine cpteqr (compz, n, d, e, z, ldz, work, info)
    CPTEQR
subroutine dpteqr (compz, n, d, e, z, ldz, work, info)
    DPTEQR
subroutine spteqr (compz, n, d, e, z, ldz, work, info)
    SPTEQR
subroutine zpteqr (compz, n, d, e, z, ldz, work, info)
    ZPTEQR

🌈 Detailed Description 🌈

okay so the detailed description is basically: the function docs are right below, and each routine gets her own little spotlight. no extra lore, just the tea. 🍓

👑 Function Documentation 👑

💖 cpteqr — CPTEQR 💖

subroutine cpteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

Purpose

so CPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor. 💅

this routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. that means if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. periodt. ✨

the eigenvectors of a full or band positive definite Hermitian matrix can also be found if CHETRD, CHPTRD, or CHBTRD has been used to reduce this matrix to tridiagonal form. 🦋

important caveat, bestie: (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.)

Parameters

COMPZ — COMPZ is CHARACTER*1

  • = 'N': Compute eigenvalues only.
  • = 'V': Compute eigenvectors of original Hermitian matrix also. Array Z contains the unitary matrix used to reduce the original matrix to tridiagonal form.
  • = 'I': Compute eigenvectors of tridiagonal matrix also.

N — N is INTEGER The order of the matrix. N >= 0.

D — D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order.

E — E is REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.

Z — Z is COMPLEX array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original Hermitian matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced.

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

WORK — WORK is REAL array, dimension (4*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 Cholesky factorization of the matrix could not be performed because the leading principal minor of order i was not positive.
    • > N the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero.

Author

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

Definition at line 144 of file cpteqr.f.

💙 dpteqr — DPTEQR 💙

subroutine dpteqr (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)

Purpose

DPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using DPTTRF, and then calling DBDSQR to compute the singular values of the bidiagonal factor. 🔥

this routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. that means if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. slay. 💜

the eigenvectors of a full or band symmetric positive definite matrix can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to reduce this matrix to tridiagonal form. 🫶

caveat time, bestie: (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.)

Parameters

COMPZ — COMPZ is CHARACTER*1

  • = 'N': Compute eigenvalues only.
  • = 'V': Compute eigenvectors of original symmetric matrix also. Array Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.
  • = 'I': Compute eigenvectors of tridiagonal matrix also.

N — N is INTEGER The order of the matrix. N >= 0.

D — D is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order.

E — E is DOUBLE PRECISION array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.

Z — Z is DOUBLE PRECISION array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original symmetric matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced.

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

WORK — WORK is DOUBLE PRECISION array, dimension (4*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 Cholesky factorization of the matrix could not be performed because the leading principal minor of order i was not positive.
    • > N the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero.

Author

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

Definition at line 144 of file dpteqr.f.

💜 spteqr — SPTEQR 💜

subroutine spteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

Purpose

SPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF, and then calling SBDSQR to compute the singular values of the bidiagonal factor. 🧚

this routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. that means if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. iconic. ⭐

the eigenvectors of a full or band symmetric positive definite matrix can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to reduce this matrix to tridiagonal form. 🌸

and yes, the same caveat applies: (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.)

Parameters

COMPZ — COMPZ is CHARACTER*1

  • = 'N': Compute eigenvalues only.
  • = 'V': Compute eigenvectors of original symmetric matrix also. Array Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.
  • = 'I': Compute eigenvectors of tridiagonal matrix also.

N — N is INTEGER The order of the matrix. N >= 0.

D

okay so continuing the spteqr parameter parade — we do not leave a list half-finished, that’s not the vibe:

  • D — a REAL array, dimension (N). On entry, it’s holding the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order. sorted and serving, periodt.
  • E — a REAL array, dimension (N-1). On entry, it has the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. rip bestie, she did her job and left no crumbs.
  • Z — a REAL array, dimension (LDZ, N). On entry, if COMPZ = 'V', this is the orthogonal matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', it’s the orthonormal eigenvectors of the original symmetric matrix; if COMPZ = 'I', it’s the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced. she’s off duty, do not perceive her.
  • LDZ — INTEGER. This is the leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', then LDZ >= max(1,N). respect the dimension rules or catch hands.
  • WORK — a REAL array, dimension (4*N). the workspace, the backstage crew, the unseen icon.
  • INFO — INTEGER. The receipts:
    • = 0: successful exit. we love a clean finish.
    • < 0: if INFO = -i, the i-th argument had an illegal value. oop, someone broke the dress code.
    • > 0: if INFO = i, and i is:
      • <= N: the Cholesky factorization of the matrix could not be performed because the leading principal minor of order i was not positive. not positive = not passing the vibe check.
      • > N: the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero. the math said “no thoughts, head empty” and refused to settle.

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 144 of file spteqr.f.

🦋 subroutine zpteqr 🦋

subroutine zpteqr (character compz, integer n, double precision, dimension(
* ) d, double precision, dimension( * ) e, complex*16, dimension( ldz, *
) z, integer ldz, double precision, dimension( * ) work, integer info)

🌈 ZPTEQR 🌈

✨ Purpose ✨

okay so ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using DPTTRF and then calling ZBDSQR to compute the singular values of the bidiagonal factor. it’s giving pipeline, it’s giving collab.

This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. This means that if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. tiny eigenvalues get their moment, no cap.

The eigenvectors of a full or band positive definite Hermitian matrix can also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to reduce this matrix to tridiagonal form. (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.) important caveat, bestie — read the fine print.

🍓 Parameters 🍓

  • COMPZ — CHARACTER*1
    • 'N': Compute eigenvalues only. minimal effort, iconic results.
    • 'V': Compute eigenvectors of original Hermitian matrix also. Array Z contains the unitary matrix used to reduce the original matrix to tridiagonal form. we’re going back to the original source.
    • 'I': Compute eigenvectors of tridiagonal matrix also. tridiagonal girlies, this one’s for you.
  • N — INTEGER. The order of the matrix. N >= 0. zero is allowed, we’re inclusive.
  • D — DOUBLE PRECISION array, dimension (N). On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order. descending order is the only order.
  • E — DOUBLE PRECISION array, dimension (N-1). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. gone, reduced to atoms.
  • Z — COMPLEX*16 array, dimension (LDZ, N). On entry, if COMPZ = 'V', this is the unitary matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', it’s the orthonormal eigenvectors of the original Hermitian matrix; if COMPZ = 'I', it’s the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced. not referenced means she was not invited to the function.
  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', then LDZ >= max(1,N). the leading dimension is the main character, respect her.
  • WORK — DOUBLE PRECISION array, dimension (4*N). workspace, but make it double precision.
  • INFO — INTEGER. The final tea:
    • = 0: successful exit. slay, no notes.
    • < 0: if INFO = -i, the i-th argument had an illegal value. somebody passed the wrong thing and we all saw it.
    • > 0: if INFO = i, and i is:
      • <= N: the Cholesky factorization of the matrix could not be performed because the leading principal minor of order i was not positive. the minor was not positive, so the factorization said “I can’t even.”
      • > N: the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero. convergence said “not today, bestie.”

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 144 of file zpteqr.f.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

💖 and that’s the whole pteqr(3) tea — source: LAPACK, version 3.12.0, page pteqr(3). the footer didn’t give us a date, so no cap, that’s all she wrote. 🫶✨

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