# NAG FL Interfacef01abf (real_​symm_​posdef_​inv)

## 1Purpose

f01abf calculates the accurate inverse of a real symmetric positive definite matrix, using a Cholesky factorization and iterative refinement.

## 2Specification

Fortran Interface
 Subroutine f01abf ( a, lda, n, b, ldb, z,
 Integer, Intent (In) :: lda, n, ldb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: a(lda,n), b(ldb,n) Real (Kind=nag_wp), Intent (Out) :: z(n)
#include <nag.h>
 void f01abf_ (double a[], const Integer *lda, const Integer *n, double b[], const Integer *ldb, double z[], Integer *ifail)
The routine may be called by the names f01abf or nagf_matop_real_symm_posdef_inv.

## 3Description

To compute the inverse $X$ of a real symmetric positive definite matrix $A$, f01abf 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 computing ${L}^{-1}$ and then the product ${L}^{-\mathrm{T}}{L}^{-1}$. The residual matrix $R=I-AX$ is 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 inverse 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: $\mathbf{a}\left({\mathbf{lda}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: the upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$. The elements of the array below the diagonal need not be set.
On exit: the lower triangle of the inverse matrix $X$ is stored in the elements of the array below the diagonal, in rows $2$ to $n+1$; ${x}_{ij}$ is stored in ${\mathbf{a}}\left(i+1,j\right)$ for $i\ge j$. The upper triangle of the original matrix is unchanged.
2: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01abf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}+1$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{b}\left({\mathbf{ldb}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the lower triangle of the inverse matrix $X$, with ${x}_{ij}$ stored in ${\mathbf{b}}\left(i,j\right)$, for $i\ge j$.
5: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f01abf is called.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{n}}$.
6: $\mathbf{z}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
7: $\mathbf{ifail}$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:
${\mathbf{ifail}}=1$
The matrix $A$ is not positive definite, possibly due to rounding errors.
${\mathbf{ifail}}=2$
The refinement process failed to converge. The matrix $A$ is ill-conditioned.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}>{\mathbf{n}}$.
On entry, ${\mathbf{ldb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=-99$
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 inverse should be correct to full machine accuracy. For a detailed error analysis see page 40 of Wilkinson and Reinsch (1971).

## 8Parallelism and Performance

f01abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01abf 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 f01abf is approximately proportional to ${n}^{3}$.

## 10Example

This example finds the inverse of the $4$ by $4$ matrix:
 $5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 .$

### 10.1Program Text

Program Text (f01abfe.f90)

### 10.2Program Data

Program Data (f01abfe.d)

### 10.3Program Results

Program Results (f01abfe.r)