SPOSVX(3)

🦋 sposvx.f(3) 🦋

okay so real quick, before anything else — this is from the Library Functions Manual, and the page is literally called sposvx.f(3). that's the vibe. that's the whole header. moving on bestie 💅

---

🎀 NAME 🎀

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sposvx.f

like. that's it. that's the name. it's the full path to the source file, no cap, and honestly she ate with the absolute-path-as-a-name energy. very main character, very "i don't do nicknames" 😭✨

---

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

• subroutine SPOSVX (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) — SPOSVX computes the solution to system of linear equations A * X = B for PO matrices

lowkey obsessed with how that one line carries so many arguments and yet has zero thoughts about pacing. she's thorough. she's committed. she shows up with nineteen parameters and a dream 🫶

subroutine SPOSVX (fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b,
    ldb, x, ldx, rcond, ferr, berr, work, iwork, info)

and what does she actually do, you ask? she computes the solution to a system of linear equations A * X = B for PO matrices. that's the whole pitch. that's the elevator speech. PO matrices, solution computed, A * X = B, periodt 🔥

---

so yeah, that's the top of the page — the name, the signature, the mission statement. the actual tea about what all those arguments mean is coming, so don't you dare close the tab on me 💜🧚

🦋 Function/Subroutine Documentation 🦋

okay bestie, here’s the whole function glow-up 💅 SPOSVX computes the solution to system of linear equations A * X = B for PO matrices. Periodt.

   subroutine SPOSVX (character fact, character uplo, integer n, integer nrhs,
       real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af,
       integer ldaf, 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)

---

💖 Purpose 💖

SPOSVX 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 and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. slay ✨

---

✨ 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. 👑
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.
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.

• = 'F': On entry, AF contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified.
• = 'N': The matrix A will be copied to AF and factored.
• = 'E': The matrix A will be equilibrated if necessary, then copied to AF 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.

A

A is REAL array, dimension (LDA,N)

On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. 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).

LDA

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

AF

AF is REAL array, dimension (LDAF,N)

If FACT = 'F', then AF 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 AF is the factored form of the equilibrated matrix diag(S)*A*diag(S).

If FACT = 'N', then AF 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 AF 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 A for the form of the equilibrated matrix).

LDAF

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

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.

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.

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.

---

🌸 Author 🌸

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

Definition at line 304 of file sposvx.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

---

okay that’s the whole page, bestie 🫶 the footer source/version moment is LAPACK Version 3.12.0 and the source file sposvx.f — iconic source-and-version behavior, no cap 💅✨🎀