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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpbtrs (f07hs)

## Purpose

nag_lapack_zpbtrs (f07hs) solves a complex Hermitian positive definite band system of linear equations with multiple right-hand sides,
 AX = B , $AX=B ,$
where A$A$ has been factorized by nag_lapack_zpbtrf (f07hr).

## Syntax

[b, info] = f07hs(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpbtrs(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpbtrs (f07hs) is used to solve a complex Hermitian positive definite band system of linear equations AX = B$AX=B$, the function must be preceded by a call to nag_lapack_zpbtrf (f07hr) which computes the Cholesky factorization of A$A$. The solution X$X$ is computed by forward and backward substitution.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = UHU$A={U}^{\mathrm{H}}U$, where U$U$ is upper triangular; the solution X$X$ is computed by solving UHY = B${U}^{\mathrm{H}}Y=B$ and then UX = Y$UX=Y$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = LLH$A=L{L}^{\mathrm{H}}$, where L$L$ is lower triangular; the solution X$X$ is computed by solving LY = B$LY=B$ and then LHX = Y${L}^{\mathrm{H}}X=Y$.

## References

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)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = UHU$A={U}^{\mathrm{H}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LLH$A=L{L}^{\mathrm{H}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix A$A$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
3:     ab(ldab, : $:$) – complex array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The Cholesky factor of A$A$, as returned by nag_lapack_zpbtrf (f07hr).
4:     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 second dimension of the array ab.
n$n$, 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.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldab ldb

### Output Parameters

1:     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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     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: kd, 4: nrhs_p, 5: ab, 6: ldab, 7: b, 8: ldb, 9: 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.

## Accuracy

For each right-hand side vector b$b$, the computed solution x$x$ is the exact solution of a perturbed system of equations (A + E)x = b$\left(A+E\right)x=b$, where
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |E|c(k + 1)ε|UH||U|$|E|\le c\left(k+1\right)\epsilon |{U}^{\mathrm{H}}||U|$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |E|c(k + 1)ε|L||LH|$|E|\le c\left(k+1\right)\epsilon |L||{L}^{\mathrm{H}}|$,
c(k + 1)$c\left(k+1\right)$ is a modest linear function of k + 1$k+1$, and ε$\epsilon$ is the machine precision
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 (‖x − x̂‖∞)/(‖x‖∞) ≤ c(k + 1)cond(A,x)ε $‖x-x^‖∞ ‖x‖∞ ≤c(k+1)cond(A,x)ε$
where cond(A,x) = |A1||A||x| / xcond(A) = |A1||A|κ(A)$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_zpbrfs (f07hv), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ ( = κ1(A)$\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_zpbcon (f07hu).

The total number of real floating point operations is approximately 16nkr$16nkr$, assuming nk$n\gg k$.
This function may be followed by a call to nag_lapack_zpbrfs (f07hv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dpbtrs (f07he).

## Example

```function nag_lapack_zpbtrs_example
uplo = 'L';
kd = int64(1);
ab = [complex(9.39),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
b = [ -12.42 + 68.42i,  54.3 - 56.56i;
-9.93 + 0.88i,  18.32 + 4.76i;
-27.3 - 0.01i,  -4.4 + 9.97i;
5.31 + 23.63i,  9.43 + 1.41i];
[ab, info] = nag_lapack_zpbtrf(uplo, kd, ab);
[bOut, info] = nag_lapack_zpbtrs(uplo, kd, ab, b)
```
```

bOut =

-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

info =

0

```
```function f07hs_example
uplo = 'L';
kd = int64(1);
ab = [complex(9.39),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
b = [ -12.42 + 68.42i,  54.3 - 56.56i;
-9.93 + 0.88i,  18.32 + 4.76i;
-27.3 - 0.01i,  -4.4 + 9.97i;
5.31 + 23.63i,  9.43 + 1.41i];
[ab, info] = f07hr(uplo, kd, ab);
[bOut, info] = f07hs(uplo, kd, ab, b)
```
```

bOut =

-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

info =

0

```