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NAG Toolbox: nag_lapack_dpptrs (f07ge)

Purpose

nag_lapack_dpptrs (f07ge) 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_dpptrf (f07gd), using packed storage.

Syntax

[b, info] = f07ge(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpptrs(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dpptrs (f07ge) 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_dpptrf (f07gd) which computes the Cholesky factorization of A$A$, using packed storage. 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:     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 Cholesky factor of A$A$ stored in packed form, as returned by nag_lapack_dpptrf (f07gd).
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 array ap and the second dimension of the array ap. (An error is raised if these dimensions are not equal.)
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: uplo, 2: n, 3: nrhs_p, 4: ap, 5: b, 6: ldb, 7: 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_dpprfs (f07gh), 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_dppcon (f07gg).

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_dpprfs (f07gh) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zpptrs (f07gs).

Example

```function nag_lapack_dpptrs_example
uplo = 'L';
ap = [2.039607805437114;
-1.529705854077835;
0.2745625891934577;
-0.04902903378454601;
1.640121946685673;
-0.2499814119483738;
0.6737303907389101;
0.7887488055748053;
0.6616575633742563;
0.5346894269298685];
b = [8.7, 8.3;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5];
[bOut, info] = nag_lapack_dpptrs(uplo, ap, b)
```
```

bOut =

1.0000    4.0000
-1.0000    3.0000
2.0000    2.0000
-3.0000    1.0000

info =

0

```
```function f07ge_example
uplo = 'L';
ap = [2.039607805437114;
-1.529705854077835;
0.2745625891934577;
-0.04902903378454601;
1.640121946685673;
-0.2499814119483738;
0.6737303907389101;
0.7887488055748053;
0.6616575633742563;
0.5346894269298685];
b = [8.7, 8.3;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5];
[bOut, info] = f07ge(uplo, ap, b)
```
```

bOut =

1.0000    4.0000
-1.0000    3.0000
2.0000    2.0000
-3.0000    1.0000

info =

0

```

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Chapter Contents
Chapter Introduction
NAG Toolbox

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