laln2(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

okay bestie, gather round 💖 this is laln2(3) from the Library Functions Manual, and it's literally two lil' subroutines — dlaln2 and slaln2 — that do a 1x1 or 2x2 solve as one step inside trevc. Same routine twice, once in double precision, once in single. no cap, that's the whole vibe.

NAME

laln2 — laln2: 1x1 or 2x2 solve, step in trevc 🌸

SYNOPSIS

✨ Functions ✨

subroutine dlaln2 (ltrans, na, nw, smin, ca, a, lda, d1, d2, b, ldb, wr,
    wi, x, ldx, scale, xnorm, info)

DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

subroutine slaln2 (ltrans, na, nw, smin, ca, a, lda, d1, d2, b, ldb, wr,
    wi, x, ldx, scale, xnorm, info)

SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

Detailed Description

lowkey there is no prose here at all 😭 the page just goes straight into the function documentation. iconic of it honestly. moving on 🫶

Function Documentation

subroutine dlaln2 (logical ltrans, integer na, integer nw, double precision smin, double precision ca, double precision, dimension( lda, * ) a, integer lda, double precision d1, double precision d2, double precision, dimension( ldb, * ) b, integer ldb, double precision wr, double precision wi, double precision, dimension( ldx, * ) x, integer ldx, double precision scale, double precision xnorm, integer info)

DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

🎀 Purpose

ok so picture this: DLALN2 solves a system of the form

(ca A - w D ) X = s B

or

(ca A**T - w D) X = s B

with possible scaling (that's the s 💅) and perturbation of A. (A**T means A-transpose, iykyk.)

  • A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or complex value, and X and B are NA x 1 matrices — real if w is real, complex if w is complex. NA may be 1 or 2.
  • If w is complex, X and B are represented as NA x 2 matrices, the first column of each being the real part and the second being the imaginary part.
  • s is a scaling factor (<= 1), computed by DLALN2, which is so chosen that X can be computed without overflow. X is further scaled if necessary to assure that norm(ca A - w D)*norm(X) is less than overflow.
  • If both singular values of (ca A - w D) are less than SMIN, SMIN*identity will be used instead of (ca A - w D). If only one singular value is less than SMIN, one element of (ca A - w D) will be perturbed enough to make the smallest singular value roughly SMIN. If both singular values are at least SMIN, (ca A - w D) will not be perturbed. In any case, the perturbation will be at most some small multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values are computed by infinity-norm approximations, and thus will only be correct to a factor of 2 or so. so like, close enough 🤷‍♀️
  • Note: all input quantities are assumed to be smaller than overflow by a reasonable factor. (See BIGNUM.)

💜 Parameters

  • LTRANS — LTRANS is LOGICAL
    • =.TRUE.: A-transpose will be used.
    • =.FALSE.: A will be used (not transposed.)
  • NA — NA is INTEGER. The size of the matrix A. It may (only) be 1 or 2. literally only those two, don't try it 🙈
  • NW — NW is INTEGER. 1 if w is real, 2 if w is complex. It may only be 1 or 2.
  • SMIN — SMIN is DOUBLE PRECISION. The desired lower bound on the singular values of A. This should be a safe distance away from underflow or overflow, say, between (underflow/machine precision) and (machine precision * overflow ). (See BIGNUM and ULP.)
  • CA — CA is DOUBLE PRECISION. The coefficient c, which A is multiplied by.
  • A — A is DOUBLE PRECISION array, dimension (LDA,NA). The NA x NA matrix A.
  • LDA — LDA is INTEGER. The leading dimension of A. It must be at least NA.
  • D1 — D1 is DOUBLE PRECISION. The 1,1 element in the diagonal matrix D.
  • D2 — D2 is DOUBLE PRECISION. The 2,2 element in the diagonal matrix D. Not used if NA=1.
  • B — B is DOUBLE PRECISION array, dimension (LDB,NW). The NA x NW matrix B (right-hand side). If NW=2 (w is complex), column 1 contains the real part of B and column 2 contains the imaginary part.
  • LDB — LDB is INTEGER. The leading dimension of B. It must be at least NA.
  • WR — WR is DOUBLE PRECISION. The real part of the scalar w.
  • WI — WI is DOUBLE PRECISION. The imaginary part of the scalar w. Not used if NW=1.
  • X — X is DOUBLE PRECISION array, dimension (LDX,NW). The NA x NW matrix X (unknowns), as computed by DLALN2. If NW=2 (w is complex), on exit, column 1 will contain the real part of X and column 2 will contain the imaginary part.
  • LDX — LDX is INTEGER. The leading dimension of X. It must be at least NA.
  • SCALE — SCALE is DOUBLE PRECISION. The scale factor that B must be multiplied by to insure that overflow does not occur when computing X. Thus, (ca A - w D) X will be SCALE*B, not B (ignoring perturbations of A.) It will be at most 1.
  • XNORM — XNORM is DOUBLE PRECISION. The infinity-norm of X, when X is regarded as an NA x NW real matrix.
  • INFO — INFO is INTEGER. An error flag. It will be set to zero if no error occurs, a negative number if an argument is in error, or a positive number if ca A - w D had to be perturbed. The possible values are:
    • = 0: No error occurred, and (ca A - w D) did not have to be perturbed.
    • = 1: (ca A - w D) had to be perturbed to make its smallest (or only) singular value greater than SMIN.
NOTE: In the interests of speed, this routine does not check the inputs for errors. 😭 she's fast, not careful, respect it

🌈 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 216 of file dlaln2.f.

subroutine slaln2 (logical ltrans, integer na, integer nw, real smin, real ca, real, dimension( lda, * ) a, integer lda, real d1, real d2, real, dimension( ldb, * ) b, integer ldb, real wr, real wi, real, dimension( ldx, * ) x, integer ldx, real scale, real xnorm, integer info)

SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

🦋 Purpose

same whole entire mood as DLALN2, just single precision this time bestie. SLALN2 solves a system of the form

(ca A - w D ) X = s B

or

(ca A**T - w D) X = s B

with possible scaling (the s again 💅) and perturbation of A. (A**T means A-transpose.)

  • A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or complex value, and X and B are NA x 1 matrices — real if w is real, complex if w is complex. NA may be 1 or 2.
  • If w is complex, X and B are represented as NA x 2 matrices, the first column of each being the real part and the second being the imaginary part.
  • s is a scaling factor (<= 1), computed by SLALN2, which is so chosen that X can be computed without overflow. X is further scaled if necessary to assure that norm(ca A - w D)*norm(X) is less than overflow.
  • If both singular values of (ca A - w D) are less than SMIN, SMIN*identity will be used instead of (ca A - w D). If only one singular value is less than SMIN, one element of (ca A - w D) will be perturbed enough to make the smallest singular value roughly SMIN. If both singular values are at least SMIN, (ca A - w D) will not be perturbed. In any case, the perturbation will be at most some small multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values are computed by infinity-norm approximations, and thus will only be correct to a factor of 2 or so.
  • Note: all input quantities are assumed to be smaller than overflow by a reasonable factor. (See BIGNUM.)

🍓 Parameters

  • LTRANS — LTRANS is LOGICAL
    • =.TRUE.: A-transpose will be used.
    • =.FALSE.: A will be used (not transposed.)
  • NA — NA is INTEGER. The size of the matrix A. It may (only) be 1 or 2.
  • NW — NW is INTEGER. 1 if w is real, 2 if w is complex. It may only be 1 or 2.
  • SMIN — SMIN is REAL. The desired lower bound on the singular values of A. This should be a safe distance away from underflow or overflow, say, between (underflow/machine precision) and (machine precision * overflow ). (See BIGNUM and ULP.)
  • CA — CA is REAL. The coefficient c, which A is multiplied by.
  • A — A is REAL array, dimension (LDA,NA). The NA x NA matrix A.
  • LDA — LDA is INTEGER. The leading dimension of A. It must be at least NA.
  • D1 — D1 is REAL. The 1,1 element in the diagonal matrix D.
  • D2 — D2 is REAL. The 2,2 element in the diagonal matrix D. Not used if NA=1.
  • B — B is REAL array, dimension (LDB,NW). The NA x NW matrix B (right-hand side). If NW=2 (w is complex), column 1 contains the real part of B and column 2 contains the imaginary part.
  • LDB — LDB is INTEGER. The leading dimension of B. It must be at least NA.
  • WR — WR is REAL. The real part of the scalar w.
  • WI — WI is REAL. The imaginary part of the scalar w. Not used if NW=1. 🧚

okay so that's where part 1 runs out of page, bestie — the rest of slaln2's params (and everything after) is waiting for you in part 2 💖✨

Okay so we're in the output-parameter era of this page and honestly it's giving season-finale energy 🎀 Everything down here is what SLALN2 hands back to you once it's done doing its thing — read these, bestie, they're the receipts.

  • WI — REAL

The imaginary part of the scalar . Not used if NW=1. (and yeah, the page literally trails off right there at "the scalar ." — we don't write 'em, we just report 'em 💀)

  • X — REAL array, dimension (LDX,NW)

The NA x NW matrix X (unknowns), as computed by SLALN2. If NW=2 ( is complex), on exit, column 1 will contain the real part of X and column 2 will contain the imaginary part. Two real columns doing the job of one complex column, we love a budget hack that slays 💅

  • LDX — INTEGER

The leading dimension of X. It must be at least NA. It must be at least NA, bestie — that's not a suggestion, that's the vibe check ✨

  • SCALE — REAL

The scale factor that B must be multiplied by to insure that overflow does not occur when computing X. Thus, (ca A - w D) X will be SCALE*B, not B (ignoring perturbations of A.) It will be at most 1.

  • XNORM — REAL

The infinity-norm of X, when X is regarded as an NA x NW real matrix.

  • INFO — INTEGER

An error flag. It will be set to zero if no error occurs, a negative number if an argument is in error, or a positive number if ca A - w D had to be perturbed. The possible values are:

  • = 0: No error occurred, and (ca A - w D) did not have to be perturbed. iconic, no notes 🌈
  • = 1: (ca A - w D) had to be perturbed to make its smallest (or only) singular value greater than SMIN. a lil nudge to keep things invertible, no cap 🫶
⚠️ NOTE: In the interests of speed, this routine does not check the inputs for errors. Translation: it trusts you completely. Betray that trust and it's on you, bestie 😭🔥

👑 AUTHOR 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Four institutions, one routine, absolute main character energy 🌸

Definition at line 216 of file slaln2.f. (yes, if you wanna go read the actual Fortran, that's where it lives — go off, researcher 🔥)

Author (the Doxygen one, not the humans)

Generated automatically by Doxygen for LAPACK from the source code. So like, a robot wrote this bit, and then a slightly chattier robot rewrote the robot's bit. It's giving… lineage 🧚

That's the whole page, bestie! 💖 Footer deets for the record books: this is from LAPACK, Version 3.12.0, with the source/version and date stamped on exactly like the manpage does it. Go forth and be the most well-conditioned version of yourself 🦋✨👑

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