# NAG FL Interfacef04abf (withdraw_​real_​posdef_​solve_​ref)

Note: this routine is deprecated and will be withdrawn at Mark 28. Replaced by f07fbf.

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## 1Purpose

f04abf calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides, using a Cholesky factorization and iterative refinement.

## 2Specification

Fortran Interface
 Subroutine f04abf ( a, lda, b, ldb, n, m, c, ldc, bb, ldbb,
 Integer, Intent (In) :: lda, ldb, n, m, ldc, ldbb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,m), bb(ldbb,m) Real (Kind=nag_wp), Intent (Out) :: wkspce(n)
#include <nag.h>
 void f04abf_ (double a[], const Integer *lda, const double b[], const Integer *ldb, const Integer *n, const Integer *m, double c[], const Integer *ldc, double wkspce[], double bb[], const Integer *ldbb, Integer *ifail)
The routine may be called by the names f04abf or nagf_linsys_withdraw_real_posdef_solve_ref.

## 3Description

Given a set of real linear equations $AX=B$, where $A$ is symmetric positive definite, f04abf first computes a Cholesky factorization of $A$ as $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular. An approximation to $X$ is found by forward and backward substitution. The residual matrix $R=B-AX$ 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.

## 4References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5Arguments

1: $\mathbf{a}\left({\mathbf{lda}},*\right)$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 upper triangle of the $n×n$ positive definite symmetric matrix $A$. The elements of the array below the diagonal need not be set.
On exit: the elements of the array below the diagonal are overwritten; the upper triangle of $A$ is unchanged.
2: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f04abf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
3: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry: the $n×m$ right-hand side matrix $B$.
4: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f04abf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{m}$Integer Input
On entry: $m$, the number of right-hand sides.
Constraint: ${\mathbf{m}}\ge 0$.
7: $\mathbf{c}\left({\mathbf{ldc}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On exit: the $n×m$ solution matrix $X$.
8: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f04abf is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{wkspce}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
10: $\mathbf{bb}\left({\mathbf{ldbb}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array bb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On exit: the final $n×m$ residual matrix $R=B-AX$.
11: $\mathbf{ldbb}$Integer Input
On entry: the first dimension of the array bb as declared in the (sub)program from which f04abf is called.
Constraint: ${\mathbf{ldbb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
Matrix $A$ is not positive definite.
${\mathbf{ifail}}=2$
Matrix $A$ is too ill-conditioned; iterative refinement fails to improve the solution.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{ldb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{ldbb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldbb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{ldc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

## 8Parallelism and Performance

f04abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04abf 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 implementation-specific information.

The time taken by f04abf is approximately proportional to ${n}^{3}$.
If there is only one right-hand side, it is simpler to use f04asf.

## 10Example

This example solves the set of linear equations $AX=B$ where
 $A=( 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 ) and B=( 23 32 33 31 ) .$

### 10.1Program Text

Program Text (f04abfe.f90)

### 10.2Program Data

Program Data (f04abfe.d)

### 10.3Program Results

Program Results (f04abfe.r)