Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_real_symm_packed_solve (f04bj)

## Purpose

nag_linsys_real_symm_packed_solve (f04bj) computes the solution to a real system of linear equations AX = B$AX=B$, where A$A$ is an n$n$ by n$n$ symmetric matrix, stored in packed format and X$X$ and B$B$ are n$n$ by r$r$ matrices. An estimate of the condition number of A$A$ and an error bound for the computed solution are also returned.

## Syntax

[ap, ipiv, b, rcond, errbnd, ifail] = f04bj(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, ipiv, b, rcond, errbnd, ifail] = nag_linsys_real_symm_packed_solve(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

The diagonal pivoting method is used to factor A$A$ as A = UDUT$A=UD{U}^{\mathrm{T}}$, if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or A = LDLT$A=LD{L}^{\mathrm{T}}$, if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where U$U$ (or L$L$) is a product of permutation and unit upper (lower) triangular matrices, and D$D$ is symmetric and block diagonal with 1$1$ by 1$1$ and 2$2$ by 2$2$ diagonal blocks. The factored form of A$A$ is then used to solve the system of equations AX = B$AX=B$.

## 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix A$A$ is stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix A$A$ is stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The n$n$ by n$n$ symmetric matrix A$A$, packed column-wise in a linear array. The j$j$th column of the matrix A$A$ is stored in the array ap as follows:
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, ap(i + (j1)j / 2) = aij${\mathbf{ap}}\left(i+\left(j-1\right)j/2\right)={a}_{ij}$, for 1ij$1\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, ap(i + (j1)(2nj) / 2) = aij${\mathbf{ap}}\left(i+\left(j-1\right)\left(2n-j\right)/2\right)={a}_{ij}$, for jin$j\le i\le n$.
See Section [Further Comments] below for further details.
3:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ matrix of right-hand sides B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
The number of linear equations n$n$, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides r$r$, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldb

### Output Parameters

1:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, the block diagonal matrix D$D$ and the multipliers used to obtain the factor U$U$ or L$L$ from the factorization A = UDUT$A=UD{U}^{\mathrm{T}}$ or A = LDLT$A=LD{L}^{\mathrm{T}}$ as computed by nag_lapack_dsptrf (f07pd), stored as a packed triangular matrix in the same storage format as A$A$.
2:     ipiv(n) – int64int32nag_int array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, details of the interchanges and the block structure of D$D$, as determined by nag_lapack_dsptrf (f07pd).
• If ipiv(k) > 0${\mathbf{ipiv}}\left(k\right)>0$, then rows and columns k$k$ and ipiv(k)${\mathbf{ipiv}}\left(k\right)$ were interchanged, and dkk${d}_{kk}$ is a 1$1$ by 1$1$ diagonal block;
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ and ipiv(k) = ipiv(k1) < 0${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k-1\right)<0$, then rows and columns k1$k-1$ and ipiv(k)$-{\mathbf{ipiv}}\left(k\right)$ were interchanged and dk1 : k,k1 : k${d}_{k-1:k,k-1:k}$ is a 2$2$ by 2$2$ diagonal block;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$ and ipiv(k) = ipiv(k + 1) < 0${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k+1\right)<0$, then rows and columns k + 1$k+1$ and ipiv(k)$-{\mathbf{ipiv}}\left(k\right)$ were interchanged and dk : k + 1,k : k + 1${d}_{k:k+1,k:k+1}$ is a 2$2$ by 2$2$ diagonal block.
3:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A$A$, computed as rcond = 1 / (A1A11)${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
5:     errbnd – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that x1 / x1errbnd${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and x$x$ is the corresponding column of the exact solution X$X$. If rcond is less than machine precision, then errbnd is returned as unity.
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail < 0andifail999${\mathbf{ifail}}<0 \text{and} {\mathbf{ifail}}\ne -999$
If ifail = i${\mathbf{ifail}}=-i$, the i$i$th argument had an illegal value.
ifail = 999${\mathbf{ifail}}=-999$
Allocation of memory failed. The integer allocatable memory required is n, and the double allocatable memory required is 2 × n$2×{\mathbf{n}}$. Allocation failed before the solution could be computed.
ifail > 0andifailN${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{N}}$
If ifail = i${\mathbf{ifail}}=i$, dii${d}_{ii}$ is exactly zero. The factorization has been completed, but the block diagonal matrix D$D$ is exactly singular, so the solution could not be computed.
W ifail = N + 1${\mathbf{ifail}}={\mathbf{N}}+1$
rcond is less than machine precision, so that the matrix A$A$ is numerically singular. A solution to the equations AX = B$AX=B$ has nevertheless been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b, $(A+E) x^=b,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε$\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A$A$ with respect to the solution of the linear equations. nag_linsys_real_symm_packed_solve (f04bj) uses the approximation E1 = εA1${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The packed storage scheme is illustrated by the following example when n = 4$n=4$ and uplo = 'U'${\mathbf{uplo}}=\text{'U'}$. Two-dimensional storage of the symmetric matrix A$A$:
 a11 a12 a13 a14 a22 a23 a24 a33 a34 a44
(aij = aji)
$a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 ( aij = aji )$
Packed storage of the upper triangle of A$A$:
ap = a11,a12,a22,a13,a23,a33,a14,a24,a34,a44
 [ ]
$ap= [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]$
The total number of floating point operations required to solve the equations AX = B$AX=B$ is proportional to ((1/3)n3 + 2n2r)$\left(\frac{1}{3}{n}^{3}+2{n}^{2}r\right)$. 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 nag_linsys_real_symm_packed_solve (f04bj) are nag_linsys_complex_herm_packed_solve (f04cj) for complex Hermitian matrices, and nag_linsys_complex_symm_packed_solve (f04dj) for complex symmetric matrices.

## Example

function nag_linsys_real_symm_packed_solve_example
uplo = 'U';
ap = [-1.81;
2.06;
1.15;
0.63;
1.87;
-0.21;
-1.15;
4.2;
3.87;
2.07];
b = [0.96, 3.93;
6.07, 19.25;
8.38, 9.9;
9.5, 27.85];
[apOut, ipiv, bOut, rcond, errbnd, ifail] = nag_linsys_real_symm_packed_solve(uplo, ap, b)

apOut =

0.4074
0.3031
-2.5907
-0.5960
0.8115
1.1500
0.6537
0.2230
4.2000
2.0700

ipiv =

1
2
-2
-2

bOut =

-5.0000    2.0000
-2.0000    3.0000
1.0000    4.0000
4.0000    1.0000

rcond =

0.0132

errbnd =

8.4029e-15

ifail =

0

function f04bj_example
uplo = 'U';
ap = [-1.81;
2.06;
1.15;
0.63;
1.87;
-0.21;
-1.15;
4.2;
3.87;
2.07];
b = [0.96, 3.93;
6.07, 19.25;
8.38, 9.9;
9.5, 27.85];
[apOut, ipiv, bOut, rcond, errbnd, ifail] = f04bj(uplo, ap, b)

apOut =

0.4074
0.3031
-2.5907
-0.5960
0.8115
1.1500
0.6537
0.2230
4.2000
2.0700

ipiv =

1
2
-2
-2

bOut =

-5.0000    2.0000
-2.0000    3.0000
1.0000    4.0000
4.0000    1.0000

rcond =

0.0132

errbnd =

8.4029e-15

ifail =

0