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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpftrs (f07we)

## Purpose

nag_lapack_dpftrs (f07we) 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_dpftrf (f07wd), stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

## Syntax

[b, info] = f07we(transr, uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpftrs(transr, uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dpftrs (f07we) is used to solve a real symmetric positive definite system of linear equations AX = B$AX=B$, the function must be preceded by a call to nag_lapack_dpftrf (f07wd) which computes the Cholesky factorization of A$A$, where A$A$ is stored in RFP format. 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

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## Parameters

### Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the RFP representation of A$A$ is normal or transposed.
transr = 'N'${\mathbf{transr}}=\text{'N'}$
The matrix A$A$ is stored in normal RFP format.
transr = 'T'${\mathbf{transr}}=\text{'T'}$
The matrix A$A$ is stored in transposed RFP format.
Constraint: transr = 'N'${\mathbf{transr}}=\text{'N'}$ or 'T'$\text{'T'}$.
2:     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'$\text{'L'}$.
3:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – double array
The Cholesky factorization of A$A$ stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).
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 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${\mathbf{nrhs}}\ge 0$.

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: transr, 2: uplo, 3: n, 4: nrhs_p, 5: a, 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)$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$ and κ(A)${\kappa }_{\infty }\left(A\right)$ is the condition number when using the $\infty$-norm.
Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$.

The total number of floating point operations is approximately 2n2r$2{n}^{2}r$.
The complex analogue of this function is nag_lapack_zpftrs (f07ws).

## Example

```function nag_lapack_dpftrs_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
b = [  8.70,  8.30;
-13.35,  2.13;
1.89,  1.61;
-4.14,  5.00];
transr = 'n';
uplo   = 'l';
n      = int64(4);
% Factorize a
[aOut, info] = nag_lapack_dpftrf(transr, uplo, n, a);

if info == 0
% Compute solution
[bOut, info] = nag_lapack_dpftrs(transr, uplo, aOut, b);
fprintf('\n');
[ifail] = nag_file_print_matrix_real_gen('g', ' ', bOut, 'Solutions');
else
fprintf('\na is not positive definite.\n');
end
```
```

Solutions
1          2
1      1.0000     4.0000
2     -1.0000     3.0000
3      2.0000     2.0000
4     -3.0000     1.0000

```
```function f07we_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
b = [  8.70,  8.30;
-13.35,  2.13;
1.89,  1.61;
-4.14,  5.00];
transr = 'n';
uplo   = 'l';
n      = int64(4);
% Factorize a
[aOut, info] = f07wd(transr, uplo, n, a);

if info == 0
% Compute solution
[bOut, info] = f07we(transr, uplo, aOut, b);
fprintf('\n');
[ifail] = x04ca('g', ' ', bOut, 'Solutions');
else
fprintf('\na is not positive definite.\n');
end
```
```

Solutions
1          2
1      1.0000     4.0000
2     -1.0000     3.0000
3      2.0000     2.0000
4     -3.0000     1.0000

```