SRC/spteqr.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💖 SRC/spteqr.f 💖

okay so this is the spteqr page from the LAPACK manual — like, the library functions manual, bestie. it's a Fortran subroutine page and honestly? it's kind of iconic. let's get into it 🎀

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

subroutine spteqr (compz, n, d, e, z, ldz, work, info)

SPTEQR — that's the whole name, she's short and she knows it 💅

🦋 Function/Subroutine Documentation 🦋

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

👑 SPTEQR 👑

🍓 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.

so she factors with SPTTRF, then hands off to SBDSQR for the singular values of the bidiagonal factor. teamwork, fr fr 🤝

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.

high relative accuracy is THE moment here ✨ like, if your eigenvalues span a huge range of magnitudes, the tiny ones and their eigenvectors come out way more accurate than with the standard QR method. not me getting emotional about numerical stability 😭

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. (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.)

bestie, read the parenthetical, it's important 💀 if you got here via SSYTRD, SSPTRD, or SSBTRD reducing your full or band symmetric positive definite matrix down to tridiagonal form, you can still get the eigenvectors — but that reduction step may lowkey ruin your shot at high relative accuracy for the small eigenvalues of the original matrix, when those eigenvalues range over many orders of magnitude. just so you know 🫶

🌈 Parameters 🌈

COMPZ

COMPZ is CHARACTER*1
  • = 'N' — eigenvalues only, that's it, no eigenvectors, we're being efficient 💅
  • = 'V' — also compute the eigenvectors of the original symmetric matrix. array Z holds the orthogonal matrix used to reduce the original matrix to tridiagonal form.
  • = 'I' — also compute the eigenvectors of the tridiagonal matrix. different vibe, same energy ✨

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.

entry: diagonals. exit: eigenvalues, biggest first, descending order, main character energy 👑🔥

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.

rip E, she does not survive the function 😭💀 destroyed on exit, no cap.

Z

Z is REAL 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.

so like — if INFO > 0 on exit, Z only holds the eigenvectors for the eigenvalues that actually got stored 🙈 and if COMPZ = 'N', Z is literally not referenced at all, she's off duty.

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)

four times N, that's the whole spec, slay 🌸

INFO

INFO is INTEGER
  • = 0 — successful exit, we love to see it 🎉
  • < 0 — if INFO = -i, the i-th argument had an illegal value. bestie you passed something wrong 😬
  • > 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. (so it's not positive definite, the vibe check failed 💀)
    • > 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 ⭐

Definition at line 144 of file spteqr.f.

🧚 Author (again, she's listed twice) 🧚

Generated automatically by Doxygen for LAPACK from the source code.

🫶 cute little sign-off: this absolute icon of a page comes from LAPACK, Version 3.12.0, for SRC/spteqr.f. go forth and factor your tridiagonal matrices, bestie 🦋✨
📦 where this page comes from