DPOSVX(3)

okay so this is DPOSVX(3) from the Library Functions Manual, and we are literally in the group chat for linear algebra, bestie. 💖

🌸 NAME 🌸

The NAME section is giving “just the source file, nothing else” energy:

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

✨ SYNOPSIS ✨

💖 Functions/Subroutines 💖

This bit is filed under Functions/Subroutines, and the main character is this subroutine right here:

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

And what does it do? DPOSVX computes the solution to system of linear equations A * X = B for PO matrices. That’s the whole vibe: PO matrices, linear system, solution unlocked. slay. 💅

✨ PURPOSE ✨

okay so bestie, here's the whole vibe of DPOSVX: it 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. 🫶

and like — bonus — error bounds on the solution and a condition estimate are also provided. it's giving overachiever and we love that for us. 💅

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🌸 DESCRIPTION 🌸

here's the lil step-by-step of what actually goes down, no cap:

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. ⚠️ (yes it warns you and then keeps going — main character energy fr)
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. ✨ glow-up era ✨
6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

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🎀 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 options are:

• = '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.

pick your fighter bestie 💜

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 DOUBLE PRECISION 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). 😭 she's changed, she's grown

LDA

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

AF

AF is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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. 💖 the payoff!!

LDX

LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).

RCOND

RCOND is DOUBLE PRECISION. 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 DOUBLE PRECISION 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. ngl that's lowkey reassuring

BERR

BERR is DOUBLE PRECISION 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 DOUBLE PRECISION 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. 💀 rip
  • = 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. (plot twist!!)

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👑 AUTHOR 👑

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

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Definition at line 304 of file dposvx.f. 📍

✨ AUTHOR ✨

okay so nobody actually wrote this page by hand bestie 😭 it was

Generated automatically by Doxygen for LAPACK from the source code.

which is lowkey iconic of Doxygen ngl — it literally read the source code and generated this whole thing for you, no cap 🫶 so if you spot something that reads a little robot-ish, that's why. it's giving "made by the machine, for the people" energy 💅

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and that's a wrap on DPOSVX(3) fr fr 🎀 you made it to the end, which is main character behavior honestly 👑

source + version vibes: this page comes from LAPACK, Version 3.12.0, generated from the source file dposvx.f ✨ treat your linear algebra with love and always check your INFO before you trust the answer 🙈💖

bye bestie, go solve something symmetric positive definite for me 🌸🦋