SRC/sppsvx.f(3)

💅 sppsvx.f — Library Functions Manual ✨

okay so bestie, this is SRC/sppsvx.f(3) from the Library Functions Manual 🎀 — and yes, we’re keeping every crumb. No cap. 🫶

🌸 NAME 🌸

SRC/sppsvx.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

• subroutine sppsvx — SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 💖. The full signature is:

subroutine sppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x,
           ldx, rcond, ferr, berr, work, iwork, info)

💅 Function/Subroutine Documentation 💅

   subroutine sppsvx (character fact, character uplo, integer n, integer nrhs,
       real, dimension( * ) ap, real, dimension( * ) afp, character equed,
       real, dimension( * ) s, real, dimension( ldb, * ) b, integer ldb, real,
       dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * )
       ferr, real, dimension( * ) berr, real, dimension( * ) work, integer,
       dimension( * ) iwork, integer info)

Okay bestie, so SPPSVX computes the solution to a system of linear equations A * X = B for OTHER matrices. Iconic. 💖

✨ Purpose ✨

SPPSVX uses the Cholesky factorization A = U**T*U or A = L*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 matrix stored in packed format and X and B are N-by-NRHS matrices. No cap, it also gives you error bounds on the solution and a condition estimate. ✨

🎀 Description 🎀

The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate the system:

   diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

      A = U**T* U,  if UPLO = 'U', or
      A = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix.

3. 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. Periodt.
4. The system of equations is solved for X using the factored form of A.
5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Obsessed with the accuracy glow-up. ✨
6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

🌸 Parameters 🌸

• FACT — FACT is CHARACTER*1. Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. The values are giving:
  • = 'F': On entry, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified.
  • = 'N': The matrix A will be copied to AFP and factored.
  • = 'E': The matrix A will be equilibrated if necessary, then copied to AFP and factored.
• UPLO — UPLO is CHARACTER*1.
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — N is INTEGER. The number of linear equations, i.e., 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.
• AP — AP is REAL array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The j-th column of A is stored in the array AP as follows:
  • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
  • if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

See below for further details. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).

• AFP — AFP is REAL array, dimension (N*(N+1)/2). If FACT = 'F', then AFP is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AFP is the factored form of the equilibrated matrix A.

If FACT = 'N', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T * U or A = L * L**T of the original matrix A.

If FACT = 'E', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T * U or A = L * L**T of the equilibrated matrix A (see the description of AP for the form of the equilibrated matrix).

• EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done.
  • = 'N': No equilibration (always true if FACT = 'N').
  • = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.

• S — S is REAL array, dimension (N). The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. Positive vibes only, literally.
• B — B is REAL array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * 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 to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
• LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
• RCOND — RCOND is REAL. The estimate of the reciprocal condition number of the matrix A after equilibration (if done). 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. Big yikes if that happens, but the routine tells you.
• FERR — FERR is 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 — 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 (3*N).
• IWORK — IWORK is INTEGER array, dimension (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. iykyk, sometimes the estimate is just not the whole story.

👑 Author 👑

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

🧚 Further Details 🧚

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

Packed storage of the upper triangle of A:

 AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 309 of file sppsvx.f.

👑 AUTHOR 👑

Okay bestie, real talk — nobody wrote this one by hand, and I need you to know that up front 💅

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and that's a wrap on sppsvx.f(3), bestie! 🎀

This iconic little artifact came to you courtesy of LAPACK, Version 3.12.0, from the source file SRC/sppsvx.f(3) 🌸 She's documented, she's factored, she's slaying. Go forth and solve those systems 💖🦋