# NAG FL Interfacef07maf (dsysv)

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## 1Purpose

f07maf computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ symmetric matrix and $X$ and $B$ are $n×r$ matrices.

## 2Specification

Fortran Interface
 Subroutine f07maf ( uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info)
 Integer, Intent (In) :: n, nrhs, lda, ldb, lwork Integer, Intent (Inout) :: ipiv(*) Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07maf_ (const char *uplo, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, Integer ipiv[], double b[], const Integer *ldb, double work[], const Integer *lwork, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07maf, nagf_lapacklin_dsysv or its LAPACK name dsysv.

## 3Description

f07maf uses the diagonal pivoting method to factor $A$ as $A=UD{U}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=LD{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, and $D$ is symmetric and block diagonal with $1×1$ and $2×2$ diagonal blocks. The factored form of $A$ is then used to solve the system of equations $AX=B$.
Note that, in general, different permutations (pivot sequences) and diagonal block structures are obtained for ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$

## 4References

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

## 5Arguments

1: $\mathbf{uplo}$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 $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
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: $\mathbf{a}\left({\mathbf{lda}},*\right)$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 $n×n$ 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 block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by f07mdf.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07maf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{ipiv}\left(*\right)$Integer array Output
Note: the dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: details of the interchanges and the block structure of $D$. More precisely,
• if ${\mathbf{ipiv}}\left(i\right)=k>0$, ${d}_{ii}$ is a $1×1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a $2×2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
• if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a $2×2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.
7: $\mathbf{b}\left({\mathbf{ldb}},*\right)$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 $n×r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n×r$ solution matrix $X$.
8: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07maf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ returns the optimal lwork.
10: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f07maf is called.
${\mathbf{lwork}}\ge 1$, and for best performance ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\mathit{nb}\right)$, where $\mathit{nb}$ is the optimal block size for f07mdf.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
11: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.
${\mathbf{info}}>0$
Element $⟨\mathit{\text{value}}⟩$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $(A+E) x^=b ,$
where
 $‖E‖1 = O(ε) ‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
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.
f07mbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bhf solves $Ax=b$ and returns a forward error bound and condition estimate. f04bhf calls f07maf to solve the equations.

## 8Parallelism and Performance

f07maf 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 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 analogues of f07maf are f07mnf for Hermitian matrices, and f07nnf for symmetric matrices.

## 10Example

This example solves the equations
 $Ax=b ,$
where $A$ is the symmetric matrix
 $A = ( -1.81 2.06 0.63 -1.15 2.06 1.15 1.87 4.20 0.63 1.87 -0.21 3.87 -1.15 4.20 3.87 2.07 ) and b = ( 0.96 6.07 8.38 9.50 ) .$
Details of the factorization of $A$ are also output.

### 10.1Program Text

Program Text (f07mafe.f90)

### 10.2Program Data

Program Data (f07mafe.d)

### 10.3Program Results

Program Results (f07mafe.r)