NAG FL Interface
f04bjf (real_symm_packed_solve)
1
Purpose
f04bjf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric matrix, stored in packed format and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n, nrhs, ldb 
Integer, Intent (Inout) 
:: 
ifail 
Integer, Intent (Out) 
:: 
ipiv(n) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
ap(*), b(ldb,*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
rcond, errbnd 
Character (1), Intent (In) 
:: 
uplo 

C Header Interface
#include <nag.h>
void 
f04bjf_ (const char *uplo, const Integer *n, const Integer *nrhs, double ap[], Integer ipiv[], double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail, const Charlen length_uplo) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f04bjf_ (const char *uplo, const Integer &n, const Integer &nrhs, double ap[], Integer ipiv[], double b[], const Integer &ldb, double &rcond, double &errbnd, Integer &ifail, const Charlen length_uplo) 
}

The routine may be called by the names f04bjf or nagf_linsys_real_symm_packed_solve.
3
Description
The diagonal pivoting method is used 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 $1$ by $1$ and $2$ by $2$ diagonal blocks. 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
https://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5
Arguments

1:
$\mathbf{uplo}$ – Character(1)
Input

On entry: if
${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix
$A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.

2:
$\mathbf{n}$ – Integer
Input

On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{nrhs}$ – Integer
Input

On entry: the number of righthand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.

4:
$\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the dimension of the array
ap
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right)$.
On entry: the
$n$ by
$n$ symmetric matrix
$A$, packed columnwise in a linear array. The
$j$th column of the matrix
$A$ is stored in the array
ap as follows:
More precisely,
 if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j1\right)/2\right)$ for $i\le j$;
 if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2nj\right)\left(j1\right)/2\right)$ for $i\ge j$.
On exit: if
${\mathbf{ifail}}\ge {\mathbf{0}}$, 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
f07pdf, stored as a packed triangular matrix in the same storage format as
$A$.

5:
$\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer array
Output

On exit: if
${\mathbf{ifail}}\ge {\mathbf{0}}$, details of the interchanges and the block structure of
$D$, as determined by
f07pdf.
 If ${\mathbf{ipiv}}\left(k\right)>0$, then rows and columns $k$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged, and ${d}_{kk}$ is a $1$ by $1$ diagonal block;
 if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k1\right)<0$, then rows and columns $k1$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k1:k,k1:k}$ is a $2$ by $2$ diagonal block;
 if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k+1\right)<0$, then rows and columns $k+1$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k:k+1,k:k+1}$ is a $2$ by $2$ diagonal block.

6:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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 $n$ by $r$ matrix of righthand sides $B$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n$ by $r$ solution matrix $X$.

7:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f04bjf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

8:
$\mathbf{rcond}$ – Real (Kind=nag_wp)
Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.

9:
$\mathbf{errbnd}$ – Real (Kind=nag_wp)
Output

On exit: if
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for a computed solution
$\hat{x}$, such that
${\Vert \hat{x}x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{errbnd}}$, where
$\hat{x}$ is a column of the computed solution returned in the array
b and
$x$ is the corresponding column of the exact solution
$X$. If
rcond is less than
machine precision,
errbnd is returned as unity.

10:
$\mathbf{ifail}$ – 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).
6
Error 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:
 ${\mathbf{ifail}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{ifail}}\le {\mathbf{n}}$

Diagonal block $\u2329\mathit{\text{value}}\u232a$ of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.
 ${\mathbf{ifail}}={\mathbf{n}}+1$

A solution has been computed, but
rcond is less than
machine precision so that the matrix
$A$ is numerically singular.
 ${\mathbf{ifail}}=1$

On entry,
uplo not one of 'U' or 'u' or 'L' or 'l':
${\mathbf{uplo}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{nrhs}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
 ${\mathbf{ifail}}=7$

On entry, ${\mathbf{ldb}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
The integer allocatable memory required is
n, and the real allocatable memory required is
$2\times {\mathbf{n}}$. Allocation failed before the solution could be computed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
The computed solution for a single righthand side,
$\hat{x}$, satisfies an equation of the form
where
and
$\epsilon $ is the
machine precision. An approximate error bound for the computed solution is given by
where
$\kappa \left(A\right)={\Vert {A}^{1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of
$A$ with respect to the solution of the linear equations.
f04bjf uses the approximation
${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f04bjf 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 implementationspecific information.
The packed storage scheme is illustrated by the following example when
$n=4$ and
${\mathbf{uplo}}=\text{'U'}$. Twodimensional storage of the symmetric matrix
$A$:
Packed storage of the upper triangle of
$A$:
The total number of floatingpoint operations required to solve the equations $AX=B$ is proportional to $\left(\frac{1}{3}{n}^{3}+2{n}^{2}r\right)$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The complex analogues of
f04bjf are
f04cjf for complex Hermitian matrices, and
f04djf for complex symmetric matrices.
10
Example
This example solves the equations
where
$A$ is the symmetric indefinite matrix
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
10.1
Program Text
10.2
Program Data
10.3
Program Results