DLALN2(3)

hey bestie 💖 welcome to the dlaln2.f(3) page from the Library Functions Manual. the file chilling in the NAME section is literally /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaln2.f — yes, the whole path, no cap. this routine is giving tiny-linear-system solver energy and we are gonna read it together 🫶

📛 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaln2.f — that's the file, bestie. that's the whole NAME section vibe.

✨ 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. 💅

🧚 Function/Subroutine Documentation 🧚

subroutine DLALN2 (...)

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 🎀

okay so the whole vibe is: DLALN2 solves a system of the form

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.)

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. ✨

the scaling factor (the source literally leaves a blank before is a scaling factor (<= 1), so we are keeping that exact energy) is <= 1, computed by DLALN2, and it 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 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.

👑 Author 👑

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

Definition at line 216 of file dlaln2.f.

👑 Author (Doxygen's little credit) 👑

Generated automatically by Doxygen for LAPACK from the source code.

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