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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgetrs (f07ae)

Purpose

nag_lapack_dgetrs (f07ae) solves a real system of linear equations with multiple right-hand sides,
 AX = B   or   ATX = B , $AX=B or ATX=B ,$
where A$A$ has been factorized by nag_lapack_dgetrf (f07ad).

Syntax

[b, info] = f07ae(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dgetrs(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgetrs (f07ae) is used to solve a real system of linear equations AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$, the function must be preceded by a call to nag_lapack_dgetrf (f07ad) which computes the LU$LU$ factorization of A$A$ as A = PLU$A=PLU$. The solution is computed by forward and backward substitution.
If trans = 'N'${\mathbf{trans}}=\text{'N'}$, the solution is computed by solving PLY = B$PLY=B$ and then UX = Y$UX=Y$.
If trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$, the solution is computed by solving UTY = B${U}^{\mathrm{T}}Y=B$ and then LTPTX = Y${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
AX = B$AX=B$ is solved for X$X$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
ATX = B${A}^{\mathrm{T}}X=B$ is solved for X$X$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
2:     a(lda, : $:$) – double array
The first dimension of the array a 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,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The LU$LU$ factorization of A$A$, as returned by nag_lapack_dgetrf (f07ad).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The pivot indices, as returned by nag_lapack_dgetrf (f07ad).
4:     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$r$ right-hand side matrix B$B$.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, b The second dimension of the arrays a, ipiv.
n$n$, 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.
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

lda ldb

Output Parameters

1:     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)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: trans, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: ipiv, 7: b, 8: ldb, 9: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

For each right-hand side vector b$b$, the computed solution x$x$ is the exact solution of a perturbed system of equations (A + E)x = b$\left(A+E\right)x=b$, where
 |E| ≤ c(n)εP|L||U| , $|E|≤c(n)εP|L||U| ,$
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 ( ‖x − x̂‖∞ )/( ‖x‖∞ ) ≤ c(n)cond(A,x)ε $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$
where cond(A,x) = |A1||A||x| / x cond(A) = |A1||A| κ (A) $\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$, and cond(AT)$\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_dgerfs (f07ah), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_lapack_dgecon (f07ag) with norm = 'I'${\mathbf{norm}}=\text{'I'}$.

The total number of floating point operations is approximately 2n2r$2{n}^{2}r$.
This function may be followed by a call to nag_lapack_dgerfs (f07ah) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zgetrs (f07as).

Example

```function nag_lapack_dgetrs_example
trans = 'N';
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];
b = [9.52, 18.47;
24.35, 2.25;
0.77, -13.28;
-6.22, -6.21];

% Factorize A
[a, ipiv, info] = nag_lapack_dgetrf(a);

% Compute Solution
[b, info] = nag_lapack_dgetrs(trans, a, ipiv, b)
```
```

b =

1.0000    3.0000
-1.0000    2.0000
3.0000    4.0000
-5.0000    1.0000

info =

0

```
```function f07ae_example
trans = 'N';
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];
b = [9.52, 18.47;
24.35, 2.25;
0.77, -13.28;
-6.22, -6.21];

% Factorize A

% Compute Solution
[b, info] = f07ae(trans, a, ipiv, b)
```
```

b =

1.0000    3.0000
-1.0000    2.0000
3.0000    4.0000
-5.0000    1.0000

info =

0

```