SSPSVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 NAME 💖

okay bestie, this is the sspsvx.f(3) page from the Library Functions Manual, and the actual NAME is this whole path right here: /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sspsvx.f. the header/footer also give us the cute truncated tag /home/abui...RC/sspsvx.f(3) — slay, we're keeping all of it. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

  • subroutine SSPSVX (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
    • the tagline is literally: SSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

🦋 Function/Subroutine Documentation 🦋

🌸 subroutine SSPSVX 🌸

subroutine SSPSVX (character fact, character uplo, integer n, integer nrhs,
    real, dimension( * ) ap, real, dimension( * ) afp, integer, dimension( *
    ) ipiv, 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)

SSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

💜 Purpose 💜

okay so SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or 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 matrix stored in packed format and X and B are N-by-NRHS matrices.

error bounds on the solution and a condition estimate are also provided. yes bestie, it does the whole glow-up: solve, estimate, and error-bound. 💖

🎀 Description 🎀

the following steps are performed:

  1. if FACT = 'N', the diagonal pivoting method is used to factor A as
    A = U * D * U**T,  if UPLO = 'U', or
    A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

  1. if some D(i,i)=0, so that D 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. not me being warned but still showing up to work 😭
  2. the system of equations is solved for X using the factored form of A.
  3. iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

🌸 Parameters 🌸

  • FACT — FACT is CHARACTER*1. specifies whether or not the factored form of A has been supplied on entry.
    • = 'F': on entry, AFP and IPIV contain the factored form of A. AP, AFP and IPIV will not be modified.
    • = 'N': the matrix A will be 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). the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
    • see below for further details. bestie, the packed-storage lore is real. 🫶
  • AFP — AFP is REAL array, dimension (N*(N+1)/2).
    • if FACT = 'F', then AFP is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as a packed triangular matrix in the same storage format as A.
    • if FACT = 'N', then AFP is an output argument and on exit contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as a packed triangular matrix in the same storage format as A.
  • IPIV — IPIV is INTEGER array, dimension (N).
    • if FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by SSPTRF.
    • if IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
    • if UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. if UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    • if FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by SSPTRF.
  • 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 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 — 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: D(i,i) is exactly zero. the factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned.
      • = N+1: D 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.
💀 warning vibes: RCOND = 0 or tiny RCOND means singular-to-working-precision energy. read INFO carefully, bestie. no cap.

👑 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 = aji)
                a44

packed storage of the upper triangle of A:

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

definition at line 274 of file sspsvx.f.

💅 Author 💅

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