SRC/sspsv.f(3)

✨ SRC/sspsv.f(3) ✨

okay bestie, so this is a page straight out of the Library Functions Manual — it's SRC/sspsv.f(3), the glow-up doc for the Fortran subroutine sspsv. buckle up, she's a linear-algebra icon 💅

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🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine sspsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)

SSPSV computes the solution to the system of linear equations A * X = B for OTHER matrices. periodt. 🔥

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🎀 Function/Subroutine Documentation 🎀

subroutine sspsv (character uplo, integer n, integer nrhs, real, dimension( * ) ap, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, integer info)

SSPSV computes the solution to the system of linear equations A * X = B for OTHER matrices. that's the whole vibe, no cap ✨

💜 Purpose:

!>
!> SSPSV computes 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.
!>
!> 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, D is symmetric and block diagonal with 1-by-1
!> and 2-by-2 diagonal blocks.  The factored form of A is then used to
!> solve the system of equations A * X = B.
!>

so basically: A is this N-by-N symmetric queen stored in packed format, and X and B are N-by-NRHS matrices. the diagonal pivoting method factors A into A = U * D * U**T when UPLO = 'U', or A = L * D * L**T when 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 🫶 then that factored form gets used to solve A * X = B. she ate and left no crumbs.

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🌸 Parameters 🌸

• UPLO — UPLO is CHARACTER*1. it's giving "which triangle are we doing":

!>          = '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. lowkey the main character number here 👑
• NRHS — NRHS is INTEGER. the number of right hand sides, i.e. the number of columns of the matrix B. NRHS >= 0. 💖
• AP — AP is REAL array, dimension (N*(N+1)/2). okay so on entry, this holds 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 like this:

!>          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 (she said, teasingly).

on exit? 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. iconic behaviour honestly ✨

• IPIV — IPIV is INTEGER array, dimension (N). all the deets on the interchanges and the block structure of D, as determined by SSPTRF. so:

!>          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 IPIV(k) > 0, rows and columns k and IPIV(k) were swapped 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 swapped and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. and if UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were swapped and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. iykyk 💅

• B — B is REAL array, dimension (LDB,NRHS). on entry, the N-by-NRHS right hand side matrix B. on exit, if INFO = 0, the N-by-NRHS solution matrix X. glow-up complete, she's transformed 💖
• LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N). no skipping this one bestie, it's a whole rule 📏
• INFO — INFO is INTEGER. the vibe check:

!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
!>                has been completed, but the block diagonal matrix D is
!>                exactly singular, so the solution could not be
!>                computed.

so = 0 means successful exit 🎉, < 0 means if INFO = -i the i-th argument had an illegal value 🙈, and > 0 means if INFO = i, D(i,i) is exactly zero — the factorization finished, but the block diagonal matrix D is exactly singular, so the solution could not be computed 😭 that's the sad ending, fr.

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👑 Author 👑

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

the academic royalty behind all of this. bow down 🫶

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🌈 Further Details: 🌈

okay this part is so satisfying, she really said "let me draw it for you":

!>
!>  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 ]
!>

see how the upper triangle just folds into that one flat array? the way that a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 reads like a little poem 🦋 (and remember aij = aji — symmetric queen behaviour, she only needs to store half her business).

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Definition at line 161 of file sspsv.f.

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💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code. (not a human, just a very diligent robot 🧚)

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and that's the whole page bestie! 🎀 this was LAPACK, Version 3.12.0, page SRC/sspsv.f(3) — go forth and solve those symmetric systems, you absolute icon 💜✨