F04AFF calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple righthand sides,
$AX=B$, with iterative refinement, where
$A$ has been factorized by
F03AEF.
To solve a set of real linear equations
$AX=B$ where
$A$ is symmetric positive definite, F04AFF must be preceded by a call to
F03AEF which computes a Cholesky factorization of
$A$ as
$A=L{L}^{\mathrm{T}}$, where
$L$ is lower triangular. An approximation to
$X$ is then found by forward and backward substitution. The residual matrix
$R=BAX$ is then calculated using
additional precision, and a correction
$D$ to
$X$ is found by solving
$L{L}^{\mathrm{T}}D=R$.
$X$ is replaced by
$X+D$, and this iterative refinement of the solution is repeated until full machine accuracy has been obtained.
 1: N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: IR – INTEGERInput
On entry: $r$, the number of righthand sides.
 3: A(LDA,N) – REAL (KIND=nag_wp) arrayInput
On entry: the upper triangle of the
$n$ by
$n$ positive definite symmetric matrix
$A$, and the subdiagonal elements of its Cholesky factor
$L$, as returned by
F03AEF.
On exit: is used as internal workspace, but is restored on exit.
 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F04AFF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 5: P(N) – REAL (KIND=nag_wp) arrayInput
On entry: the reciprocals of the diagonal elements of
$L$, as returned by
F03AEF.
 6: B(LDB,IR) – REAL (KIND=nag_wp) arrayInput
On entry: the $n$ by $r$ righthand side matrix $B$.
 7: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F04AFF is called.
Constraint:
${\mathbf{LDB}}\ge {\mathbf{N}}$.
 8: EPS – REAL (KIND=nag_wp)Input
On entry: must be set to the value of the machine precision.
 9: X(LDX,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ by $r$ solution matrix $X$.
 10: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F04AFF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{N}}$.
 11: BB(LDBB,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the final $n$ by $r$ residual matrix $R=BAX$.
 12: LDBB – INTEGERInput
On entry: the first dimension of the array
BB as declared in the (sub)program from which F04AFF is called.
Constraint:
${\mathbf{LDBB}}\ge {\mathbf{N}}$.
 13: K – INTEGEROutput
On exit: the number of iterations needed in the refinement process.
 14: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 39 of
Wilkinson and Reinsch (1971).
This example solves the set of linear equations
$AX=B$ where