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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

```