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# NAG Toolbox: nag_lapack_dpotrs (f07fe)

## Purpose

nag_lapack_dpotrs (f07fe) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
 AX = B , $AX=B ,$
where A$A$ has been factorized by nag_lapack_dpotrf (f07fd).

## Syntax

[b, info] = f07fe(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpotrs(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dpotrs (f07fe) is used to solve a real symmetric positive definite system of linear equations AX = B$AX=B$, this function must be preceded by a call to nag_lapack_dpotrf (f07fd) which computes the Cholesky factorization of A$A$. The solution X$X$ is computed by forward and backward substitution.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = UTU$A={U}^{\mathrm{T}}U$, where U$U$ is upper triangular; the solution X$X$ is computed by solving UTY = B${U}^{\mathrm{T}}Y=B$ and then UX = Y$UX=Y$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = LLT$A=L{L}^{\mathrm{T}}$, where L$L$ is lower triangular; the solution X$X$ is computed by solving LY = B$LY=B$ and then LTX = Y${L}^{\mathrm{T}}X=Y$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = UTU$A={U}^{\mathrm{T}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LLT$A=L{L}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
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 Cholesky factor of A$A$, as returned by nag_lapack_dpotrf (f07fd).
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$r$ 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 array a.
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: uplo, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: b, 7: ldb, 8: 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
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |E|c(n)ε|UT||U|$|E|\le c\left(n\right)\epsilon |{U}^{\mathrm{T}}||U|$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |E|c(n)ε|L||LT|$|E|\le c\left(n\right)\epsilon |L||{L}^{\mathrm{T}}|$,
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| / xcond(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)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_dporfs (f07fh), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ ( = κ1(A)$\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_dpocon (f07fg).

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_dporfs (f07fh) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zpotrs (f07fs).

## Example

```function nag_lapack_dpotrs_example
uplo = 'L';
a = [4.16, 0, 0, 0;
-3.12, 5.03, 0, 0;
0.56, -0.83, 0.76, 0;
-0.1, 1.18, 0.34, 1.18];
b = [8.7, 8.3;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5];
[a, info] = nag_lapack_dpotrf(uplo, a);
[bOut, info] = nag_lapack_dpotrs(uplo, a, b)
```
```

bOut =

1.0000    4.0000
-1.0000    3.0000
2.0000    2.0000
-3.0000    1.0000

info =

0

```
```function f07fe_example
uplo = 'L';
a = [4.16, 0, 0, 0;
-3.12, 5.03, 0, 0;
0.56, -0.83, 0.76, 0;
-0.1, 1.18, 0.34, 1.18];
b = [8.7, 8.3;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5];
[a, info] = f07fd(uplo, a);
[bOut, info] = f07fe(uplo, a, b)
```
```

bOut =

1.0000    4.0000
-1.0000    3.0000
2.0000    2.0000
-3.0000    1.0000

info =

0

```

Chapter Contents
Chapter Introduction
NAG Toolbox

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