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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dppsv (f07ga)

## Purpose

nag_lapack_dppsv (f07ga) computes the solution to a real system of linear equations
 AX = B , $AX=B ,$
where A$A$ is an n$n$ by n$n$ symmetric positive definite matrix stored in packed format and X$X$ and B$B$ are n$n$ by r$r$ matrices.

## Syntax

[ap, b, info] = f07ga(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, b, info] = nag_lapack_dppsv(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dppsv (f07ga) uses the Cholesky decomposition to factor A$A$ as A = UTU$A={U}^{\mathrm{T}}U$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = LLT$A=L{L}^{\mathrm{T}}$ if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where U$U$ is an upper triangular matrix and L$L$ is a lower triangular matrix. The factored form of A$A$ is then used to solve the system of equations AX = B$AX=B$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ is stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ is stored.
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 n$n$ by n$n$ symmetric matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.
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 b.
n$n$, the number of linear equations, i.e., 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, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldb

### Output Parameters

1:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the factor U$U$ or L$L$ from the Cholesky factorization A = UTU$A={U}^{\mathrm{T}}U$ or A = LLT$A=L{L}^{\mathrm{T}}$, in the same storage format as A$A$.
2:     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)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the n$n$ by r$r$ solution matrix X$X$.
3:     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.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the leading minor of order i$i$ of A$A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b , $(A+E) x^=b ,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
nag_lapack_dppsvx (f07gb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_packed_solve (f04be) solves Ax = b $Ax=b$ and returns a forward error bound and condition estimate. nag_linsys_real_posdef_packed_solve (f04be) calls nag_lapack_dppsv (f07ga) to solve the equations.

The total number of floating point operations is approximately (1/3) n3 + 2n2r $\frac{1}{3}{n}^{3}+2{n}^{2}r$, where r $r$ is the number of right-hand sides.
The complex analogue of this function is nag_lapack_zppsv (f07gn).

## Example

```function nag_lapack_dppsv_example
uplo = 'U';
ap = [4.16;
-3.12;
5.03;
0.56;
-0.83;
0.76;
-0.1;
1.18;
0.34;
1.18];
b = [8.7;
-13.35;
1.89;
-4.14];
[apOut, bOut, info] = nag_lapack_dppsv(uplo, ap, b)
```
```

apOut =

2.0396
-1.5297
1.6401
0.2746
-0.2500
0.7887
-0.0490
0.6737
0.6617
0.5347

bOut =

1.0000
-1.0000
2.0000
-3.0000

info =

0

```
```function f07ga_example
uplo = 'U';
ap = [4.16;
-3.12;
5.03;
0.56;
-0.83;
0.76;
-0.1;
1.18;
0.34;
1.18];
b = [8.7;
-13.35;
1.89;
-4.14];
[apOut, bOut, info] = f07ga(uplo, ap, b)
```
```

apOut =

2.0396
-1.5297
1.6401
0.2746
-0.2500
0.7887
-0.0490
0.6737
0.6617
0.5347

bOut =

1.0000
-1.0000
2.0000
-3.0000

info =

0

```