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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zspsvx (f07qp)

## Purpose

nag_lapack_zspsvx (f07qp) uses the diagonal pivoting factorization
 A = UDUT   or   A = LDLT $A=UDUT or A=LDLT$
to compute 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. Error bounds on the solution and a condition estimate are also provided.

## Syntax

[afp, ipiv, x, rcond, ferr, berr, info] = f07qp(fact, uplo, ap, afp, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[afp, ipiv, x, rcond, ferr, berr, info] = nag_lapack_zspsvx(fact, uplo, ap, afp, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zspsvx (f07qp) performs the following steps:
1. If fact = 'N'${\mathbf{fact}}=\text{'N'}$, the diagonal pivoting method is used 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 and D$D$ is symmetric and block diagonal with 1$1$ by 1$1$ and 2$2$ by 2$2$ diagonal blocks.
2. If some dii = 0${d}_{ii}=0$, so that D$D$ is exactly singular, then the function returns with info = i${\mathbf{info}}=i$. Otherwise, the factored form of A$A$ is used to estimate the condition number of the matrix A$A$. If the reciprocal of the condition number is less than machine precision, infon + 1${\mathbf{info}}\ge {\mathbf{n}}+1$ is returned as a warning, but the function still goes on to solve for X$X$ and compute error bounds as described below.
3. The system of equations is solved for X$X$ using the factored form of A$A$.
4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

## 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:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix A$A$ has been supplied.
fact = 'F'${\mathbf{fact}}=\text{'F'}$
afp and ipiv contain the factorized form of the matrix A$A$. afp and ipiv will not be modified.
fact = 'N'${\mathbf{fact}}=\text{'N'}$
The matrix A$A$ will be copied to afp and factorized.
Constraint: fact = 'F'${\mathbf{fact}}=\text{'F'}$ or 'N'$\text{'N'}$.
2:     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'$\text{'L'}$.
3:     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$i\ge j$.
4:     afp( : $:$) – complex array
Note: the dimension of the array afp 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 fact = 'F'${\mathbf{fact}}=\text{'F'}$, afp contains 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$A$.
5:     ipiv(n) – int64int32nag_int array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
If fact = 'F'${\mathbf{fact}}=\text{'F'}$, ipiv contains details of the interchanges and the block structure of D$D$, as determined by nag_lapack_zsptrf (f07qr).
• 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.
6:     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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array ipiv.
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${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldb ldx work rwork

### Output Parameters

1:     afp( : $:$) – complex array
Note: the dimension of the array afp 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 fact = 'N'${\mathbf{fact}}=\text{'N'}$, afp contains 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
If fact = 'N'${\mathbf{fact}}=\text{'N'}$, ipiv contains details of the interchanges and the block structure of D$D$, as determined by nag_lapack_zsptrf (f07qr), as described above.
3:     x(ldx, : $:$) – complex array
The first dimension of the array x 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     rcond – double scalar
The estimate of the reciprocal condition number of the matrix A$A$. If rcond = 0.0${\mathbf{rcond}}=0.0$, the matrix may be exactly singular. This condition is indicated by INFO > 0andINFOn${\mathbf{INFO}}>{\mathbf{0}} \text{and} {\mathbf{INFO}}\le \mathbf{n}$. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by infon + 1${\mathbf{info}}\ge {\mathbf{n}}+1$.
5:     ferr(nrhs_p) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that jxj / xjferr(j)${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$ where j${\stackrel{^}{x}}_{j}$ is the j$j$th column of the computed solution returned in the array x and xj${x}_{j}$ is the corresponding column of the exact solution X$X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
6:     berr(nrhs_p) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the component-wise relative backward error of each computed solution vector j${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of A$A$ or B$B$ that makes j${\stackrel{^}{x}}_{j}$ an exact solution).
7:     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: fact, 2: uplo, 3: n, 4: nrhs_p, 5: ap, 6: afp, 7: ipiv, 8: b, 9: ldb, 10: x, 11: ldx, 12: rcond, 13: ferr, 14: berr, 15: work, 16: rwork, 17: 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 > 0andINFON${\mathbf{INFO}}>0 \text{and} {\mathbf{INFO}}\le {\mathbf{N}}$
If ${\mathbf{info}}\le {\mathbf{n}}$, d(i,i)$d\left(i,i\right)$ is exactly zero. The factorization has been completed, but the factor D$D$ is exactly singular, so the solution and error bounds could not be computed. rcond = 0.0${\mathbf{rcond}}=0.0$ is returned.
W INFO = N + 1${\mathbf{INFO}}={\mathbf{N}}+1$
D$D$ is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

## Accuracy

For each right-hand side vector b$b$, the computed solution $\stackrel{^}{x}$ is the exact solution of a perturbed system of equations (A + E) = b$\left(A+E\right)\stackrel{^}{x}=b$, where
 ‖E‖1 = O(ε) ‖A‖1 , $‖E‖1 = O(ε) ‖A‖1 ,$
where ε$\epsilon$ is the machine precision. See Chapter 11 of Higham (2002) for further details.
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 ( ‖x − x̂‖∞ )/( ‖x̂‖∞ ) ≤ wc cond(A,x̂,b) $‖x-x^‖∞ ‖x^‖∞ ≤ wc cond(A,x^,b)$
where cond(A,,b) = |A1|(|A||| + |b|) / cond(A) = |A1||A|κ (A)$\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖|{A}^{-1}|\left(|A||\stackrel{^}{x}|+|b|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the j $j$th column of X $X$, then wc ${w}_{c}$ is returned in berr(j) ${\mathbf{berr}}\left(j\right)$ and a bound on x / ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ferr(j) ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

The factorization of A $A$ requires approximately (4/3) n3 $\frac{4}{3}{n}^{3}$ floating point operations.
For each right-hand side, computation of the backward error involves a minimum of 16n2 $16{n}^{2}$ floating point operations. Each step of iterative refinement involves an additional 24n2 $24{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form Ax = b $Ax=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 8n2 $8{n}^{2}$ operations.
The real analogue of this function is nag_lapack_dspsvx (f07pb). The complex Hermitian analogue of this function is nag_lapack_zhpsvx (f07pp).

## Example

```function nag_lapack_zspsvx_example
fact = 'Not factored';
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];
afp = complex(zeros(10, 1));
ipiv = [int64(0);0;0;0];
b = [ -6.43 + 19.24i,  -4.59 - 35.53i;
-0.49 - 1.47i,  6.95 + 20.49i;
-48.18 + 66i,  -12.08 - 27.02i;
-55.64 + 41.22i,  -19.09 - 35.97i];
[afpOut, ipivOut, x, rcond, ferr, berr, info] = nag_lapack_zspsvx(fact, uplo, ap, afp, ipiv, b)
```
```

afpOut =

-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

ipivOut =

1
2
-2
-2

x =

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

rcond =

0.0486

ferr =

1.0e-13 *

0.1190
0.1254

berr =

1.0e-16 *

0.8207
0.7349

info =

0

```
```function f07qp_example
fact = 'Not factored';
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];
afp = complex(zeros(10, 1));
ipiv = [int64(0);0;0;0];
b = [ -6.43 + 19.24i,  -4.59 - 35.53i;
-0.49 - 1.47i,  6.95 + 20.49i;
-48.18 + 66i,  -12.08 - 27.02i;
-55.64 + 41.22i,  -19.09 - 35.97i];
[afpOut, ipivOut, x, rcond, ferr, berr, info] = f07qp(fact, uplo, ap, afp, ipiv, b)
```
```

afpOut =

-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

ipivOut =

1
2
-2
-2

x =

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

rcond =

0.0486

ferr =

1.0e-13 *

0.1190
0.1254

berr =

1.0e-16 *

0.8207
0.7349

info =

0

```