SRC/sptsvx.f(3)

Ok so this is SRC/sptsvx.f(3), and the manual it lives in is the Library Functions Manual. Yes bestie, we're doing Fortran linear algebra and we're gonna make it make sense. 💅✨

💖 NAME

SRC/sptsvx.f — that's the file, bestie. This page is about the sptsvx subroutine, and the manual name is Library Functions Manual (title: SRC/sptsvx.f(3)). 🎀

✨ SYNOPSIS

🧚 Functions/Subroutines

subroutine sptsvx (fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond,
    ferr, berr, work, info)
     SPTSVX computes the solution to system of linear equations A * X =
    B for PT matrices

SPTSVX computes the solution to system of linear equations A * X = B for PT matrices. 🫶

🫶 Function/Subroutine Documentation

Here's the full signature, no details skipped, because we are not about that vague life:

   subroutine sptsvx (character fact, integer n, integer nrhs, real, dimension(
       * ) d, real, dimension( * ) e, real, dimension( * ) df, real, dimension(
       * ) ef, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx,
       * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real,
       dimension( * ) berr, real, dimension( * ) work, integer info)
        SPTSVX computes the solution to system of linear equations A * X = B
       for PT matrices

🌸 Purpose

SPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. 💖

Error bounds on the solution and a condition estimate are also provided. No cap.

🦋 Description

The following steps are performed:

1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**T*D*U. ✨
2. If the leading principal minor of order i is not positive, 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. Slay.
4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Obsessed with the thoroughness.

🎀 Parameters

• FACT — FACT is CHARACTER*1. Specifies whether or not the factored form of A has been supplied on entry. = 'F': On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified. = 'N': The matrix A will be copied to DF and EF and factored. 💅
• N — N is INTEGER. The order of the matrix A. N >= 0.
• NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
• D — D is REAL array, dimension (N). The n diagonal elements of the tridiagonal matrix A. 🌸
• E — E is REAL array, dimension (N-1). The (n-1) subdiagonal elements of the tridiagonal matrix A.
• DF — DF is REAL array, dimension (N). If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A. If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A.
• EF — EF is REAL array, dimension (N-1). If FACT = 'F', then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. 🦋
• B — B is REAL array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
• LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• X — X is REAL array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 👑
• LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
• RCOND — RCOND is REAL. 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 — FERR is REAL array, dimension (NRHS). The 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). 💜
• BERR — BERR is 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 — WORK is REAL array, dimension (2*N). 🍓
• INFO — INFO is INTEGER.
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: if INFO = i, and i is:
    • <= N: the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been 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. 🔥

👑 Author

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

Definition at line 226 of file sptsvx.f. 🎯

✨ Author

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That's the whole vibe, bestie! 🎀 This page comes from SRC/sptsvx.f(3) and it's LAPACK Version 3.12.0. The footer doesn't give us a date, so no invented date — just the version and the source. 💜✨