NAG FL Interface
f07fbf (dposvx)
1
Purpose
f07fbf uses the Cholesky factorization
to compute the solution to a real system of linear equations
where
$A$ is an
$n$ by
$n$ symmetric positive definite matrix and
$X$ and
$B$ are
$n$ by
$r$ matrices. Error bounds on the solution and a condition estimate are also provided.
2
Specification
Fortran Interface
Subroutine f07fbf ( 
fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) 
Integer, Intent (In) 
:: 
n, nrhs, lda, ldaf, ldb, ldx 
Integer, Intent (Out) 
:: 
iwork(n), info 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), af(ldaf,*), s(*), b(ldb,*), x(ldx,*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
rcond, ferr(nrhs), berr(nrhs), work(3*n) 
Character (1), Intent (In) 
:: 
fact, uplo 
Character (1), Intent (InOut) 
:: 
equed 

C Header Interface
#include <nag.h>
void 
f07fbf_ (const char *fact, const char *uplo, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double af[], const Integer *ldaf, char *equed, double s[], double b[], const Integer *ldb, double x[], const Integer *ldx, double *rcond, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_fact, const Charlen length_uplo, const Charlen length_equed) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f07fbf_ (const char *fact, const char *uplo, const Integer &n, const Integer &nrhs, double a[], const Integer &lda, double af[], const Integer &ldaf, char *equed, double s[], double b[], const Integer &ldb, double x[], const Integer &ldx, double &rcond, double ferr[], double berr[], double work[], Integer iwork[], Integer &info, const Charlen length_fact, const Charlen length_uplo, const Charlen length_equed) 
}

The routine may be called by the names f07fbf, nagf_lapacklin_dposvx or its LAPACK name dposvx.
3
Description
f07fbf performs the following steps:

1.If ${\mathbf{fact}}=\text{'E'}$, real diagonal scaling factors, ${D}_{S}$, are computed to equilibrate the system:
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 ${D}_{S}A{D}_{S}$ and $B$ by ${D}_{S}B$.

2.If ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, the Cholesky decomposition is used to factor the matrix $A$ (after equilibration if ${\mathbf{fact}}=\text{'E'}$) as $A={U}^{\mathrm{T}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix.

3.If the leading $i$ by $i$ principal minor of $A$ is not positive definite, then the routine returns with ${\mathbf{info}}=i$. 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, ${\mathbf{info}}={\mathbf{n}+{\mathbf{1}}}$ 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 to calculate error bounds and backward error estimates for it.

6.If equilibration was used, the matrix $X$ is premultiplied by ${D}_{S}$ so that it solves the original system before equilibration.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5
Arguments

1:
$\mathbf{fact}$ – Character(1)
Input

On entry: specifies whether or not the factorized form of the matrix
$A$ is supplied on entry, and if not, whether the matrix
$A$ should be equilibrated before it is factorized.
 ${\mathbf{fact}}=\text{'F'}$
 af contains the factorized form of $A$. If ${\mathbf{equed}}=\text{'Y'}$, the matrix $A$ has been equilibrated with scaling factors given by s. a and af will not be modified.
 ${\mathbf{fact}}=\text{'N'}$
 The matrix $A$ will be copied to af and factorized.
 ${\mathbf{fact}}=\text{'E'}$
 The matrix $A$ will be equilibrated if necessary, then copied to af and factorized.
Constraint:
${\mathbf{fact}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.

2:
$\mathbf{uplo}$ – Character(1)
Input

On entry: if
${\mathbf{uplo}}=\text{'U'}$, the upper triangle of
$A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.

3:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

4:
$\mathbf{nrhs}$ – Integer
Input

On entry: $r$, the number of righthand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.

5:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the
$n$ by
$n$ symmetric matrix
$A$.
If
${\mathbf{fact}}=\text{'F'}$ and
${\mathbf{equed}}=\text{'Y'}$,
a must have been equilibrated by the scaling factor in
s as
${D}_{S}A{D}_{S}$.
 If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
 If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
${\mathbf{fact}}=\text{'F'}$ or
$\text{'N'}$, or if
${\mathbf{fact}}=\text{'E'}$ and
${\mathbf{equed}}=\text{'N'}$,
a is not modified.
If
${\mathbf{fact}}=\text{'E'}$ and
${\mathbf{equed}}=\text{'Y'}$,
a is overwritten by
${D}_{S}A{D}_{S}$.

6:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f07fbf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

7:
$\mathbf{af}\left({\mathbf{ldaf}},*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
af
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if
${\mathbf{fact}}=\text{'F'}$,
af contains the triangular factor
$U$ or
$L$ from the Cholesky factorization
$A={U}^{\mathrm{T}}U$ or
$A=L{L}^{\mathrm{T}}$, in the same storage format as
a. If
${\mathbf{equed}}\ne \text{'N'}$,
af is the factorized form of the equilibrated matrix
${D}_{S}A{D}_{S}$.
On exit: if
${\mathbf{fact}}=\text{'N'}$,
af returns the triangular factor
$U$ or
$L$ from the Cholesky factorization
$A={U}^{\mathrm{T}}U$ or
$A=L{L}^{\mathrm{T}}$ of the original matrix
$A$.
If
${\mathbf{fact}}=\text{'E'}$,
af returns the triangular factor
$U$ or
$L$ from the Cholesky factorization
$A={U}^{\mathrm{T}}U$ or
$A=L{L}^{\mathrm{T}}$ of the equilibrated matrix
$A$ (see the description of
a for the form of the equilibrated matrix).

8:
$\mathbf{ldaf}$ – Integer
Input

On entry: the first dimension of the array
af as declared in the (sub)program from which
f07fbf is called.
Constraint:
${\mathbf{ldaf}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

9:
$\mathbf{equed}$ – Character(1)
Input/Output

On entry: if
${\mathbf{fact}}=\text{'N'}$ or
$\text{'E'}$,
equed need not be set.
If
${\mathbf{fact}}=\text{'F'}$,
equed must specify the form of the equilibration that was performed as follows:
 if ${\mathbf{equed}}=\text{'N'}$, no equilibration;
 if ${\mathbf{equed}}=\text{'Y'}$, equilibration was performed, i.e., $A$ has been replaced by ${D}_{S}A{D}_{S}$.
On exit: if
${\mathbf{fact}}=\text{'F'}$,
equed is unchanged from entry.
Otherwise, if no constraints are violated,
equed specifies the form of the equilibration that was performed as specified above.
Constraint:
if ${\mathbf{fact}}=\text{'F'}$, ${\mathbf{equed}}=\text{'N'}$ or $\text{'Y'}$.

10:
$\mathbf{s}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the dimension of the array
s
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if
${\mathbf{fact}}=\text{'N'}$ or
$\text{'E'}$,
s need not be set.
If
${\mathbf{fact}}=\text{'F'}$ and
${\mathbf{equed}}=\text{'Y'}$,
s must contain the scale factors,
${D}_{S}$, for
$A$; each element of
s must be positive.
On exit: if
${\mathbf{fact}}=\text{'F'}$,
s is unchanged from entry.
Otherwise, if no constraints are violated and
${\mathbf{equed}}=\text{'Y'}$,
s contains the scale factors,
${D}_{S}$, for
$A$; each element of
s is positive.

11:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ righthand side matrix $B$.
On exit: if
${\mathbf{equed}}=\text{'N'}$,
b is not modified.
If
${\mathbf{equed}}=\text{'Y'}$,
b is overwritten by
${D}_{S}B$.

12:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f07fbf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

13:
$\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Real (Kind=nag_wp) array
Output

Note: the second dimension of the array
x
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n$ by $r$ solution matrix $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if ${\mathbf{equed}}=\text{'Y'}$, and the solution to the equilibrated system is ${D}_{S}^{1}X$.

14:
$\mathbf{ldx}$ – Integer
Input

On entry: the first dimension of the array
x as declared in the (sub)program from which
f07fbf is called.
Constraint:
${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

15:
$\mathbf{rcond}$ – Real (Kind=nag_wp)
Output

On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as ${\mathbf{rcond}}=1.0/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.

16:
$\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: if
${\mathbf{info}}={\mathbf{0}}$ or
${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for each computed solution vector, such that
${\Vert {\hat{x}}_{j}{x}_{j}\Vert}_{\infty}/{\Vert {x}_{j}\Vert}_{\infty}\le {\mathbf{ferr}}\left(j\right)$ where
${\hat{x}}_{j}$ is the
$j$th column of the computed solution returned in the array
x and
${x}_{j}$ is the corresponding column of the exact solution
$X$. The estimate is as reliable as the estimate for
rcond, and is almost always a slight overestimate of the true error.

17:
$\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the componentwise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).

18:
$\mathbf{work}\left(3\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Workspace


19:
$\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer array
Workspace


20:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 ${\mathbf{info}}<0$
If ${\mathbf{info}}=i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
 ${\mathbf{info}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le {\mathbf{n}}$

The leading minor of order $\u2329\mathit{\text{value}}\u232a$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed. ${\mathbf{rcond}}=0.0$ is returned.
 ${\mathbf{info}}={\mathbf{n}}+1$

$U$ (or
$L$) 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.
7
Accuracy
For each righthand side vector
$b$, the computed solution
$x$ is the exact solution of a perturbed system of equations
$\left(A+E\right)x=b$, where
 if ${\mathbf{uplo}}=\text{'U'}$, $\leftE\right\le c\left(n\right)\epsilon \left{U}^{\mathrm{T}}\right\leftU\right$;
 if ${\mathbf{uplo}}=\text{'L'}$, $\leftE\right\le c\left(n\right)\epsilon \leftL\right\left{L}^{\mathrm{T}}\right$,
$c\left(n\right)$ is a modest linear function of
$n$, and
$\epsilon $ is the
machine precision. See Section 10.1 of
Higham (2002) for further details.
If
$\hat{x}$ is the true solution, then the computed solution
$x$ satisfies a forward error bound of the form
where
$\mathrm{cond}\left(A,\hat{x},b\right)={\Vert \left{A}^{1}\right\left(\leftA\right\left\hat{x}\right+\leftb\right\right)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left{A}^{1}\right\leftA\right\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
If
$\hat{x}$ is the
$j$th column of
$X$, then
${w}_{c}$ is returned in
${\mathbf{berr}}\left(j\right)$ and a bound on
${\Vert x\hat{x}\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}$ is returned in
${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f07fbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The factorization of $A$ requires approximately $\frac{1}{3}{n}^{3}$ floatingpoint operations.
For each righthand side, computation of the backward error involves a minimum of $4{n}^{2}$ floatingpoint operations. Each step of iterative refinement involves an additional $6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ operations.
The complex analogue of this routine is
f07fpf.
10
Example
This example solves the equations
where
$A$ is the symmetric positive definite matrix
and
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix $A$ are also output.
10.1
Program Text
10.2
Program Data
10.3
Program Results