SRC/dlaln2.f(3)

🌸 NAME 🌸

SRC/dlaln2.f — yes bestie, that's the name, and this page is from the Library Functions Manual. It's giving "niche LAPACK routine but make it iconic" energy. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

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.

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🫶 Function/Subroutine Documentation 🫶

Okay, here's the full signature, character for character, because she is long and she is serving:

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. No cap. 🦋

Purpose 💖

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 () and perturbation of A. (A**T means A-transpose.) Periodt. ✨

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. Very "two columns, one vibe." 🎀

There's 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. She's approximate but she's doing her best. 💀

⭐ 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). Pick your fighter. 💅
• NA — NA is INTEGER. The size of the matrix A. It may (only) be 1 or 2.
• NW — NW is INTEGER. 1 if the scalar is real, 2 if the scalar 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 (the scalar 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 (the scalar 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. Bestie, that is the whole warning. It said what it said. 😭

Author 🫶

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

Definition at line 216 of file dlaln2.f.

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🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code.

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Signed off with LAPACK Version 3.12.0 and SRC/dlaln2.f(3), periodt. 💅✨