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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zpbsvx (f07hp)

Purpose

nag_lapack_zpbsvx (f07hp) uses the Cholesky factorization
 $A=UHU or A=LLH$
to compute the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ Hermitian positive definite band matrix of bandwidth $\left(2{k}_{d}+1\right)$ and $X$ and $B$ are $n$ by $r$ matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

[ab, afb, equed, s, b, x, rcond, ferr, berr, info] = f07hp(fact, uplo, kd, ab, afb, equed, s, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, afb, equed, s, b, x, rcond, ferr, berr, info] = nag_lapack_zpbsvx(fact, uplo, kd, ab, afb, equed, s, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zpbsvx (f07hp) performs the following steps:
1. If ${\mathbf{fact}}=\text{'E'}$, real diagonal scaling factors, ${D}_{S}$, are computed to equilibrate the system:
 $DS A DS DS-1 X = DS B .$
Whether or not the system will be equilibrated depends on the scaling of the matrix $A$, but if equilibration is used, $A$ is overwritten by ${D}_{S}A{D}_{S}$ and $B$ by ${D}_{S}B$.
2. If ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, the Cholesky decomposition is used to factor the matrix $A$ (after equilibration if ${\mathbf{fact}}=\text{'E'}$) as $A={U}^{\mathrm{H}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{H}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix.
3. If the leading $i$ by $i$ principal minor of $A$ is not positive definite, then the function returns with ${\mathbf{info}}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, ${\mathbf{info}}\ge {\mathbf{n}}+1$ is returned as a warning, but the function still goes on to solve for $X$ and compute error bounds as described below.
4. The system of equations is solved for $X$ using the factored form of $A$.
5. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
6. If equilibration was used, the matrix $X$ is premultiplied by ${D}_{S}$ so that it solves the original system before equilibration.

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:     $\mathrm{fact}$ – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix $A$ is supplied on entry, and if not, whether the matrix $A$ should be equilibrated before it is factorized.
${\mathbf{fact}}=\text{'F'}$
afb contains the factorized form of $A$. If ${\mathbf{equed}}=\text{'Y'}$, the matrix $A$ has been equilibrated with scaling factors given by s. ab and afb will not be modified.
${\mathbf{fact}}=\text{'N'}$
The matrix $A$ will be copied to afb and factorized.
${\mathbf{fact}}=\text{'E'}$
The matrix $A$ will be equilibrated if necessary, then copied to afb and factorized.
Constraint: ${\mathbf{fact}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.
2:     $\mathrm{uplo}$ – string (length ≥ 1)
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{kd}$int64int32nag_int scalar
${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
4:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab must be at least ${\mathbf{kd}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of the Hermitian band matrix $A$, except if ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'Y'}$, in which case ab must contain the equilibrated matrix ${D}_{S}A{D}_{S}$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
5:     $\mathrm{afb}\left(\mathit{ldafb},:\right)$ – complex array
The first dimension of the array afb must be at least ${\mathbf{kd}}+1$.
The second dimension of the array afb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{fact}}=\text{'F'}$, afb contains the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ of the band matrix $A$, in the same storage format as $A$. If ${\mathbf{equed}}=\text{'Y'}$, afb is the factorized form of the equilibrated matrix $A$.
6:     $\mathrm{equed}$ – string (length ≥ 1)
If ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, equed need not be set.
If ${\mathbf{fact}}=\text{'F'}$, equed must specify the form of the equilibration that was performed as follows:
• if ${\mathbf{equed}}=\text{'N'}$, no equilibration;
• if ${\mathbf{equed}}=\text{'Y'}$, equilibration was performed, i.e., $A$ has been replaced by ${D}_{S}A{D}_{S}$.
Constraint: if ${\mathbf{fact}}=\text{'F'}$, ${\mathbf{equed}}=\text{'N'}$ or $\text{'Y'}$.
7:     $\mathrm{s}\left(:\right)$ – double array
The dimension of the array s must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, s need not be set.
If ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'Y'}$, s must contain the scale factors, ${D}_{S}$, for $A$; each element of s must be positive.
8:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ right-hand side matrix $B$.

Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b and the second dimension of the arrays ab, afb, s.
$n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the array b.
$r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

Output Parameters

1:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab will be ${\mathbf{kd}}+1$.
The second dimension of the array ab will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{fact}}=\text{'E'}$ and ${\mathbf{equed}}=\text{'Y'}$, ab stores ${D}_{S}A{D}_{S}$.
2:     $\mathrm{afb}\left(\mathit{ldafb},:\right)$ – complex array
The first dimension of the array afb will be ${\mathbf{kd}}+1$.
The second dimension of the array afb will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{fact}}=\text{'N'}$, afb returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$.
If ${\mathbf{fact}}=\text{'E'}$, afb returns the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ of the equilibrated matrix $A$ (see the description of ab for the form of the equilibrated matrix).
3:     $\mathrm{equed}$ – string (length ≥ 1)
If ${\mathbf{fact}}=\text{'F'}$, equed is unchanged from entry.
Otherwise, if no constraints are violated, equed specifies the form of the equilibration that was performed as specified above.
4:     $\mathrm{s}\left(:\right)$ – double array
The dimension of the array s will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{fact}}=\text{'F'}$, s is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{equed}}=\text{'Y'}$, s contains the scale factors, ${D}_{S}$, for $A$; each element of s is positive.
5:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{equed}}=\text{'N'}$, b is not modified.
If ${\mathbf{equed}}=\text{'Y'}$, b stores ${D}_{S}B$.
6:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – complex array
The first dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if ${\mathbf{equed}}=\text{'Y'}$, and the solution to the equilibrated system is ${D}_{S}^{-1}X$.
7:     $\mathrm{rcond}$ – double scalar
If no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as ${\mathbf{rcond}}=1.0/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
8:     $\mathrm{ferr}\left({\mathbf{nrhs_p}}\right)$ – double array
If ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$ where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
9:     $\mathrm{berr}\left({\mathbf{nrhs_p}}\right)$ – double array
If ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
10:   $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see 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.

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0 \text{and} {\mathbf{info}}\le {\mathbf{n}}$
The leading minor of order $_$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed. ${\mathbf{rcond}}=0.0$ is returned.
W  ${\mathbf{info}}={\mathbf{n}}+1$
$U$ (or $L$) 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 $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{H}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{H}}\right|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 10.1 of Higham (2002) for further details.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x^∞ ≤ wc condA,x^,b$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖\left|{A}^{-1}\right|\left(\left|A\right|\left|\stackrel{^}{x}\right|+\left|b\right|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left(j\right)$ and a bound on ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

When $n\gg k$, the factorization of $A$ requires approximately $4n{\left(k+1\right)}^{2}$ floating-point operations, where $k$ is the number of superdiagonals.
For each right-hand side, computation of the backward error involves a minimum of $32nk$ floating-point operations. Each step of iterative refinement involves an additional $48nk$ 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 $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $16nk$ operations.
The real analogue of this function is nag_lapack_dpbsvx (f07hb).

Example

This example solves the equations
 $AX=B ,$
where $A$ is the Hermitian positive definite band matrix
 $A = 9.39i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00$
and
 $B = -12.42+68.42i 54.30-56.56i -9.93+00.88i 18.32+04.76i -27.30-00.01i -4.40+09.97i 5.31+23.63i 9.43+01.41i .$
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix $A$ are also output.
```function f07hp_example

fprintf('f07hp example results\n\n');

uplo = 'U';
kd = int64(1);
n  = int64(4);
ab = [0,          1.08 - 1.73i,  -0.04 + 0.29i,  -0.33 + 2.24i;
9.39 + 0i,  1.69 + 0i,      2.65 + 0i,      2.17 + 0i];

% Input parameters
fact  = 'Equilibration';
afb   = complex(zeros(kd+1, n));
equed = ' ';
s     = zeros(n,1);

% RHS
b = [-12.42 + 68.42i,  54.30 - 56.56i;
-9.93 +  0.88i,  18.32 +  4.76i;
-27.30 -  0.01i,  -4.40 +  9.97i;
5.31 + 23.63i,   9.43 +  1.41i];

% Solve
[ab, afb, equed, s, b, x, rcond, ferr, berr, info] = ...
f07hp( ...
fact, uplo, kd, ab, afb, equed, s, b);

disp('Solution(s)');
disp(x);
disp('Backward errors (machine-dependent)');
fprintf('%10.1e',berr);
fprintf('\n');
disp('Estimated forward error bounds (machine-dependent)');
fprintf('%10.1e',ferr);
fprintf('\n\n');
disp('Estimate of reciprocal condition number');
fprintf('%10.1e\n\n',rcond);
if equed=='N'
fprintf('A has not been equilibrated\n');
else
fprintf('A has been equilibrated\n');
end

```
```f07hp example results

Solution(s)
-1.0000 + 8.0000i   5.0000 - 6.0000i
2.0000 - 3.0000i   2.0000 + 3.0000i
-4.0000 - 5.0000i  -8.0000 + 4.0000i
7.0000 + 6.0000i  -1.0000 - 7.0000i

Backward errors (machine-dependent)
9.1e-17   9.0e-17
Estimated forward error bounds (machine-dependent)
3.7e-14   3.2e-14

Estimate of reciprocal condition number
7.6e-03

A has not been equilibrated
```