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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

```