SRC/cgtsvx.f(3)

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version 3.12.1 · 4 versions
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💅 SRC/cgtsvx.f(3) — Library Functions Manual, but make it make sense

okay so bestie, buckle in 🎀 this is a page from the Library Functions Manual, and it's about one (1) extremely specific Fortran subroutine. she's giving linear algebra main character energy and honestly? iconic.

✨ NAME ✨

SRC/cgtsvx.f — no cap, that's literally it, that's the name 💖

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine cgtsvx (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2,
    ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
     CGTSVX computes the solution to system of linear equations A * X =
    B for GT matrices

like… look at that argument list. she's long. she's thorough. she ate 💅

🦋 Function/Subroutine Documentation 🦋

subroutine cgtsvx (character fact, character trans, integer n, integer nrhs,
    complex, dimension( * ) dl, complex, dimension( * ) d, complex,
    dimension( * ) du, complex, dimension( * ) dlf, complex, dimension( * )
    df, complex, dimension( * ) duf, complex, dimension( * ) du2, integer,
    dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb,
    complex, dimension( ldx, * ) x, integer ldx, real rcond, real,
    dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * )
    work, real, dimension( * ) rwork, integer info)

CGTSVX computes the solution to system of linear equations A * X = B for GT matrices — that's the whole vibe, fr fr 🌈

💜 Purpose 💜

okay so picture this 👑:

CGTSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a tridiagonal matrix of order N and X and B are N-by-NRHS matrices.

and that's not even the best part — she ALSO hooks you up with error bounds on the solution and a condition estimate. like, she solved it AND she's telling you how much to trust it. thoughtful queen 🫶

🔥 Description 🔥

here's the step-by-step, bestie — the routine does all of this, in order:

  1. If FACT = 'N', the LU decomposition is used to factor the matrix A as A = L * U, where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
  2. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
  3. The system of equations is solved for X using the factored form of A.
  4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
ngl step 2 is the drama of the whole page 😭 a warning but she keeps going anyway. resilient.

💖 Parameters 💖

  • FACT — CHARACTER*1. Specifies whether or not the factored form of A has been supplied on entry.
    • = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not be modified.
    • = 'N': The matrix will be copied to DLF, DF, and DUF and factored.
  • TRANS — CHARACTER*1. Specifies the form of the system of equations:
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Conjugate transpose)
  • N — INTEGER. The order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • DL — COMPLEX array, dimension (N-1). The (n-1) subdiagonal elements of A.
  • D — COMPLEX array, dimension (N). The n diagonal elements of A.
  • DU — COMPLEX array, dimension (N-1). The (n-1) superdiagonal elements of A.
  • DLF — COMPLEX array, dimension (N-1).
    • If FACT = 'F', then DLF is an input argument and on entry contains the (n-1) multipliers that define the matrix L from the LU factorization of A as computed by CGTTRF.
    • If FACT = 'N', then DLF is an output argument and on exit contains the (n-1) multipliers that define the matrix L from the LU factorization of A.
  • DF — COMPLEX array, dimension (N).
    • If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
    • If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
  • DUF — COMPLEX array, dimension (N-1).
    • If FACT = 'F', then DUF is an input argument and on entry contains the (n-1) elements of the first superdiagonal of U.
    • If FACT = 'N', then DUF is an output argument and on exit contains the (n-1) elements of the first superdiagonal of U.
  • DU2 — COMPLEX array, dimension (N-2).
    • If FACT = 'F', then DU2 is an input argument and on entry contains the (n-2) elements of the second superdiagonal of U.
    • If FACT = 'N', then DU2 is an output argument and on exit contains the (n-2) elements of the second superdiagonal of U.
  • IPIV — INTEGER array, dimension (N).
    • If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the LU factorization of A as computed by CGTTRF.
    • If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the LU factorization of A; row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required.
  • B — COMPLEX array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — COMPLEX array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — REAL. The estimate of the reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.
  • FERR — REAL array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
  • BERR — REAL array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
  • WORK — COMPLEX array, dimension (2*N).
  • RWORK — REAL array, dimension (N).
  • INFO — INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, and i is
      • <= N: U(i,i) is exactly zero. The factorization has not been completed unless i = N, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
      • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.
bestie please actually read the INFO > 0 section 💀 "singular to working precision" is not a vibe, it's a warning.

🧚 Author 🧚

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

⭐ Definition at line 291 of file cgtsvx.f. ⭐

🍓 Author 🍓

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🌟 Sign-off 🌟

and that's the whole page, bestie 🎀 — this beauty is from LAPACK, Version 3.12.0, and she lives at SRC/cgtsvx.f(3). periodt 💅✨

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