DPPSVX(3)

🎀 NAME 🎀

Manual name: dppsvx.f(3) — from the Library Functions Manual, bestie. The official NAME entry is literally this source-path situation:

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

No cap, that's the whole NAME vibe. 💅

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✨ 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

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

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🦋 Function/Subroutine Documentation 🦋

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 here's the tea: 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. Literally so helpful. ✨

🦋 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, 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 bestie, so the packed storage scheme is literally illustrated by this example when N = 4 and UPLO = 'U' — no cap:

Two-dimensional storage of the symmetric matrix A:

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

and packed storage of the upper triangle of A is giving:

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

Definition at line 309 of file dppsvx.f.

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🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole page bestie 💅 this iconic gem is from LAPACK Version 3.12.0, source file /home/abui...RC/dppsvx.f(3) — slay and goodbye 🫶✨