F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07FAF (DPOSV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07FAF (DPOSV) computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric positive definite matrix and $X$ and $B$ are $n$ by $r$ matrices.

## 2  Specification

 SUBROUTINE F07FAF ( UPLO, N, NRHS, A, LDA, B, LDB, INFO)
 INTEGER N, NRHS, LDA, LDB, INFO REAL (KIND=nag_wp) A(LDA,*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dposv.

## 3  Description

F07FAF (DPOSV) uses the Cholesky decomposition to factor $A$ 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. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## 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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     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 $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/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 ${\mathbf{UPLO}}=\text{'U'}$, 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 ${\mathbf{INFO}}={\mathbf{0}}$, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07FAF (DPOSV) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
6:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/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 $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07FAF (DPOSV) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the leading minor of order $i$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
F07FBF (DPOSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04BDF solves $Ax=b$ and returns a forward error bound and condition estimate. F04BDF calls F07FAF (DPOSV) to solve the equations.

## 8  Further Comments

The total number of floating point operations is approximately $\frac{1}{3}{n}^{3}+2{n}^{2}r$, where $r$ is the number of right-hand sides.
The complex analogue of this routine is F07FNF (ZPOSV).

## 9  Example

This example solves the equations
 $Ax=b ,$
where $A$ is the symmetric positive definite matrix
 $A = 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 and b = 8.70 -13.35 1.89 -4.14 .$
Details of the Cholesky factorization of $A$ are also output.

### 9.1  Program Text

Program Text (f07fafe.f90)

### 9.2  Program Data

Program Data (f07fafe.d)

### 9.3  Program Results

Program Results (f07fafe.r)