SRC/dppsvx.f(3)

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version 3.12.1 · 7 versions
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🎀 SRC/dppsvx.f(3) — Library Functions Manual 🎀

✨ NAME ✨

SRC/dppsvx.f — that's the file, bestie 💅

🦋 SYNOPSIS 🦋

💫 Functions/Subroutines

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

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

🌸 Function/Subroutine Documentation 🌸

👑 subroutine dppsvx — main character energy, no cap 💖

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

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

💖 Purpose 💖

Okay so DPPSVX 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 🫶

Error bounds on the solution and a condition estimate are also provided. Iconic behavior only ✨

🎀 Description 🎀

The following steps are performed, bestie — buckle up:

  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.

  1. 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.

  1. 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 🔥
  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.
  4. 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, 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.

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.

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.
      = 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.

🌈 Further Details 🌈

okay so bestie, this next bit is the lil picture that makes packed storage click — the docs literally walk you through it, no cap. Here's the situation: the packed storage scheme is illustrated by the following example when N = 4 and UPLO = 'U':

First up, the two-dimensional storage of the symmetric matrix A — this is the "normal" rectangle-of-numbers version you already know and love:

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

See how the upper triangle is the main character here? The lower-left is just vibes (it's the conjugate of the upper part, hence aij = conjg(aji)), so it's literally redundant. 🫶

And now the glow-up — packed storage of the upper triangle of A, which smooshes all of it into one flat little list:

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

Column by column, no zeros, no wasted slots, absolutely zero filler. It's giving efficiency. ✨

and for the detail-obsessed besties (me, I'm the detail-obsessed bestie): this whole routine lives at Definition at line 309 of file dppsvx.f.

🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. Not a human, not a teen girl, just a doc generator doing its thing. 💅

that's the whole page, bestie — you ate that read. 🍓 This iconic lil number comes to you from LAPACK Version 3.12.0, source SRC/dppsvx.f(3), and it is the moment for packed symmetric solves. Slay, stay curious, and go read your AP array with love. 💖🦋🌸

📦 where this page comes from