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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zptrfs (f07jv)

## Purpose

nag_lapack_zptrfs (f07jv) computes error bounds and refines the solution to a complex system of linear equations AX = B $AX=B$, where A $A$ is an n $n$ by n $n$ Hermitian positive definite tridiagonal matrix and X $X$ and B $B$ are n $n$ by r $r$ matrices, using the modified Cholesky factorization returned by nag_lapack_zpttrf (f07jr) and an initial solution returned by nag_lapack_zpttrs (f07js). Iterative refinement is used to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07jv(uplo, d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zptrfs(uplo, d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zptrfs (f07jv) should normally be preceded by calls to nag_lapack_zpttrf (f07jr) and nag_lapack_zpttrs (f07js). nag_lapack_zpttrf (f07jr) computes a modified Cholesky factorization of the matrix A $A$ as
 A = LDLH , $A=LDLH ,$
where L $L$ is a unit lower bidiagonal matrix and D $D$ is a diagonal matrix, with positive diagonal elements. nag_lapack_zpttrs (f07js) then utilizes the factorization to compute a solution, $\stackrel{^}{X}$, to the required equations. Letting $\stackrel{^}{x}$ denote a column of $\stackrel{^}{X}$, nag_lapack_zptrfs (f07jv) computes a component-wise backward error, β $\beta$, the smallest relative perturbation in each element of A $A$ and b $b$ such that $\stackrel{^}{x}$ is the exact solution of a perturbed system
 (A + E) x̂ = b + f , with  |eij| ≤ β |aij| , and  |fj| ≤ β |bj| . $(A+E) x^ = b + f , with |eij| ≤ β |aij| , and |fj| ≤ β |bj| .$
The function also estimates a bound for the component-wise forward error in the computed solution defined by max |xixi^| / max |xi^| $\mathrm{max}|{x}_{i}-\stackrel{^}{{x}_{i}}|/\mathrm{max}|\stackrel{^}{{x}_{i}}|$, where x $x$ is the corresponding column of the exact solution, X $X$.
Note that the modified Cholesky factorization of A $A$ can also be expressed as
 A = UHDU , $A=UHDU ,$
where U $U$ is unit upper bidiagonal.

## 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

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies the form of the factorization as follows:
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = UHDU$A={U}^{\mathrm{H}}DU$.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LDLH$A=LD{L}^{\mathrm{H}}$.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the matrix of A$A$.
3:     e( : $:$) – complex array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, e must contain the (n1)$\left(n-1\right)$ superdiagonal elements of the matrix A$A$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, e must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the matrix A$A$.
4:     df( : $:$) – double array
Note: the dimension of the array df must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the diagonal matrix D$D$ from the LDLT$LD{L}^{\mathrm{T}}$ factorization of A$A$.
5:     ef( : $:$) – complex array
Note: the dimension of the array ef must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, ef must contain the (n1)$\left(n-1\right)$ superdiagonal elements of the unit upper bidiagonal matrix U$U$ from the UHDU${U}^{\mathrm{H}}DU$ factorization of A$A$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, ef must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix L$L$ from the LDLH$LD{L}^{\mathrm{H}}$ factorization of A$A$.
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$r$ matrix of right-hand sides B$B$.
7:     x(ldx, : $:$) – complex array
The first dimension of the array x 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$r$ initial solution matrix X$X$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays b, x The dimension of the arrays d, df.
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 arrays b, x.
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:     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)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ refined solution matrix X$X$.
2:     ferr(nrhs_p) – double array
Estimate of the forward error bound for each computed solution vector, such that jxj / jferr(j)${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{\stackrel{^}{x}}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$, 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 almost always a slight overestimate of the true error.
3:     berr(nrhs_p) – double array
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).
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

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: d, 5: e, 6: df, 7: ef, 8: b, 9: ldb, 10: x, 11: ldx, 12: ferr, 13: berr, 14: work, 15: rwork, 16: 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

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‖∞ = O(ε)‖A‖∞ $‖E‖∞=O(ε)‖A‖∞$
and ε $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖∞)/(‖x‖∞) ≤ κ(A) (‖E‖∞)/(‖A‖∞) , $‖ x^ - x ‖∞ ‖x‖∞ ≤ κ(A) ‖E‖∞ ‖A‖∞ ,$
where κ(A) = A1 A $\kappa \left(A\right)={‖{A}^{-1}‖}_{\infty }{‖A‖}_{\infty }$, the condition number of A $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Function nag_lapack_zptcon (f07ju) can be used to compute the condition number of A $A$.

The total number of floating point operations required to solve the equations AX = B $AX=B$ is proportional to nr $nr$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The real analogue of this function is nag_lapack_dptrfs (f07jh).

## Example

```function nag_lapack_zptrfs_example
uplo = 'Lower';
d = [16;
41;
46;
21];
e = [ 16 + 16i;
18 - 9i;
1 - 4i];
df = [16;
9;
1;
4];
ef = [ 1 + 1i;
2 - 1i;
1 - 4i];
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];
x = [ 2 + 1i,  -3 - 2i;
1 + 1i,  1 + 1i;
1 - 2i,  1 - 2i;
1 - 1i,  2 + 1i];
[xOut, ferr, berr, info] = nag_lapack_zptrfs(uplo, d, e, df, ef, b, x)
```
```

xOut =

2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

ferr =

1.0e-11 *

0.9038
0.6093

berr =

0
0

info =

0

```
```function f07jv_example
uplo = 'Lower';
d = [16;
41;
46;
21];
e = [ 16 + 16i;
18 - 9i;
1 - 4i];
df = [16;
9;
1;
4];
ef = [ 1 + 1i;
2 - 1i;
1 - 4i];
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];
x = [ 2 + 1i,  -3 - 2i;
1 + 1i,  1 + 1i;
1 - 2i,  1 - 2i;
1 - 1i,  2 + 1i];
[xOut, ferr, berr, info] = f07jv(uplo, d, e, df, ef, b, x)
```
```

xOut =

2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

ferr =

1.0e-11 *

0.9038
0.6093

berr =

0
0

info =

0

```