SRC/dpteqr.f(3)

🎀 SRC/dpteqr.f(3) 🎀

Okay bestie, welcome to the SRC/dpteqr.f(3) page from the Library Functions Manual. This is the manual name, and yes, it’s giving “math girlie with a tridiagonal glow-up.” 💅

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💖 NAME 💖

SRC/dpteqr.f — literally the page name, bestie. No cap.

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• subroutine dpteqr (compz, n, d, e, z, ldz, work, info) — DPTEQR

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🌸 Function/Subroutine Documentation 🌸

Here’s the full signature, copied exactly because we do not play with code:

       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)

DPTEQR — that’s the icon’s name, periodt.

🦋 Purpose 🦋

So DPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix. The way it does this: first it factors the matrix using DPTTRF, and then it calls DBDSQR to compute the singular values of the bidiagonal factor. Slay. ✨

This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. What that means in group-chat terms: 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. It’s giving precision, honestly. 💖

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. ⚠️ But plot twist: 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 — this one is CHARACTER*1, and it decides what we’re computing:
  • = '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. It’s 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. 💀 Like, gone-gone.
• 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. Not even a little bit. 🙈
• 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. We love a clean 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.

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🤖 Author 🤖

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