SRC/slaln2.f(3)

💖 NAME 💖

SRC/slaln2.f

Okay so this page is SRC/slaln2.f(3) from the Library Functions Manual, and it’s the whole cute-but-serious scoop on slaln2. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

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. periodt. 🎀

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🧚 Function/Subroutine Documentation 🧚

🦋 subroutine slaln2 🦋

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. no cap, that’s the whole main-character job. 👑

🎀 Purpose 🎀

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 (that’s the s in the equation) 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 is <= 1, computed by SLALN2, and 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.
• X — X is REAL array, dimension (LDX,NW)
  • The NA x NW matrix X (unknowns), as computed by SLALN2. 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 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 — XNORM is REAL
  • 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 slaln2.f. 🎀

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🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code.

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That’s the whole page, bestie 🫶. Footer vibes: LAPACK Version 3.12.0, page SRC/slaln2.f(3). Mwah 💋✨