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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zspsv (f07qn)

## Purpose

nag_lapack_zspsv (f07qn) computes the solution to a complex system of linear equations
 AX = B , $AX=B ,$
where A$A$ is an n$n$ by n$n$ symmetric matrix stored in packed format and X$X$ and B$B$ are n$n$ by r$r$ matrices.

## Syntax

[ap, ipiv, b, info] = f07qn(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, ipiv, b, info] = nag_lapack_zspsv(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zspsv (f07qn) uses the diagonal pivoting method to factor A$A$ as A = UDUT$A=UD{U}^{\mathrm{T}}$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = LDLT$A=LD{L}^{\mathrm{T}}$ if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where U$U$ (or L$L$) is a product of permutation and unit upper (lower) triangular matrices, D$D$ is symmetric and block diagonal with 1$1$ by 1$1$ and 2$2$ by 2$2$ diagonal blocks. 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 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( : $:$) – complex 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, : $:$) – complex 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
Note: to solve the equations Ax = b$Ax=b$, where b$b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)$\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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( : $:$) – complex 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 block diagonal matrix D$D$ and the multipliers used to obtain the factor U$U$ or L$L$ from the factorization A = UDUT$A=UD{U}^{\mathrm{T}}$ or A = LDLT$A=LD{L}^{\mathrm{T}}$ as computed by nag_lapack_zsptrf (f07qr), stored as a packed triangular matrix in the same storage format as A$A$.
2:     ipiv(n) – int64int32nag_int array
Details of the interchanges and the block structure of D$D$. More precisely,
• if ipiv(i) = k > 0${\mathbf{ipiv}}\left(i\right)=k>0$, dii${d}_{ii}$ is a 1$1$ by 1$1$ pivot block and the i$i$th row and column of A$A$ were interchanged with the k$k$th row and column;
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ and ipiv(i1) = ipiv(i) = l < 0${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$,  ( di − 1,i − 1 di,i − 1 ) di,i − 1 dii
$\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a 2$2$ by 2$2$ pivot block and the (i1)$\left(i-1\right)$th row and column of A$A$ were interchanged with the l$l$th row and column;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$ and ipiv(i) = ipiv(i + 1) = m < 0${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=-m<0$,  ( dii di + 1,i ) di + 1,i di + 1,i + 1
$\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a 2$2$ by 2$2$ pivot block and the (i + 1)$\left(i+1\right)$th row and column of A$A$ were interchanged with the m$m$th row and column.
3:     b(ldb, : $:$) – complex 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
Note: to solve the equations Ax = b$Ax=b$, where b$b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)$\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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$X$.
4:     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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: ipiv, 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.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, dii${d}_{ii}$ is exactly zero. The factorization has been completed, but the block diagonal matrix D$D$ is exactly singular, so the solution could not be 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) and Chapter 11 of Higham (2002) for further details.
nag_lapack_zspsvx (f07qp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_symm_packed_solve (f04dj) solves AX = B $AX=B$ and returns a forward error bound and condition estimate. nag_linsys_complex_symm_packed_solve (f04dj) calls nag_lapack_zspsv (f07qn) to solve the equations.

The total number of floating point operations is approximately (4/3) n3 + 8n2r $\frac{4}{3}{n}^{3}+8{n}^{2}r$, where r $r$ is the number of right-hand sides.
The real analogue of this function is nag_lapack_dspsv (f07pa). The complex Hermitian analogue of this function is nag_lapack_zhpsv (f07pn).

## Example

```function nag_lapack_zspsv_example
uplo = 'U';
ap = [ -0.56 + 0.12i;
-1.54 - 2.86i;
-2.83 - 0.03i;
5.32 - 1.59i;
-3.52 + 0.58i;
8.86 + 1.81i;
3.8 + 0.92i;
-7.86 - 2.96i;
5.14 - 0.64i;
-0.39 - 0.71i];
b = [ -6.43 + 19.24i;
-0.49 - 1.47i;
-48.18 + 66i;
-55.64 + 41.22i];
[apOut, ipiv, bOut, info] = nag_lapack_zspsv(uplo, ap, b)
```
```

apOut =

-2.0954 - 2.2011i
-0.1071 - 0.3157i
4.4079 + 5.3991i
-0.4823 + 0.0150i
-0.6078 + 0.2811i
-2.8300 - 0.0300i
0.4426 + 0.1936i
0.5279 - 0.3715i
-7.8600 - 2.9600i
-0.3900 - 0.7100i

ipiv =

1
2
-2
-2

bOut =

-4.0000 + 3.0000i
3.0000 - 2.0000i
-2.0000 + 5.0000i
1.0000 - 1.0000i

info =

0

```
```function f07qn_example
uplo = 'U';
ap = [ -0.56 + 0.12i;
-1.54 - 2.86i;
-2.83 - 0.03i;
5.32 - 1.59i;
-3.52 + 0.58i;
8.86 + 1.81i;
3.8 + 0.92i;
-7.86 - 2.96i;
5.14 - 0.64i;
-0.39 - 0.71i];
b = [ -6.43 + 19.24i;
-0.49 - 1.47i;
-48.18 + 66i;
-55.64 + 41.22i];
[apOut, ipiv, bOut, info] = f07qn(uplo, ap, b)
```
```

apOut =

-2.0954 - 2.2011i
-0.1071 - 0.3157i
4.4079 + 5.3991i
-0.4823 + 0.0150i
-0.6078 + 0.2811i
-2.8300 - 0.0300i
0.4426 + 0.1936i
0.5279 - 0.3715i
-7.8600 - 2.9600i
-0.3900 - 0.7100i

ipiv =

1
2
-2
-2

bOut =

-4.0000 + 3.0000i
3.0000 - 2.0000i
-2.0000 + 5.0000i
1.0000 - 1.0000i

info =

0

```