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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dptrfs (f07jh)

Purpose

nag_lapack_dptrfs (f07jh) computes error bounds and refines the solution to a real system of linear equations AX = B $AX=B$, where A $A$ is an n $n$ by n $n$ symmetric 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_dpttrf (f07jd) and an initial solution returned by nag_lapack_dpttrs (f07je). Iterative refinement is used to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07jh(d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dptrfs(d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dptrfs (f07jh) should normally be preceded by calls to nag_lapack_dpttrf (f07jd) and nag_lapack_dpttrs (f07je). nag_lapack_dpttrf (f07jd) computes a modified Cholesky factorization of the matrix A $A$ as
 A = LDLT , $A=LDLT ,$
where L $L$ is a unit lower bidiagonal matrix and D $D$ is a diagonal matrix, with positive diagonal elements. nag_lapack_dpttrs (f07je) 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_dptrfs (f07jh) 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 = UTDU , $A=UTDU ,$
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:     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$A$.
2:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the matrix A$A$.
3:     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$A$.
4:     ef( : $:$) – double array
Note: the dimension of the array ef must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the unit bidiagonal matrix L$L$ from the LDLT$LD{L}^{\mathrm{T}}$ factorization of A$A$.
5:     b(ldb, : $:$) – double 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$B$.
6:     x(ldx, : $:$) – double 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, ef.
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${\mathbf{nrhs}}\ge 0$.

ldb ldx work

Output Parameters

1:     x(ldx, : $:$) – double 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: n, 2: nrhs_p, 3: d, 4: e, 5: df, 6: ef, 7: b, 8: ldb, 9: x, 10: ldx, 11: ferr, 12: berr, 13: work, 14: 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_dptcon (f07jg) 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 complex analogue of this function is nag_lapack_zptrfs (f07jv).

Example

```function nag_lapack_dptrfs_example
d = [4;
10;
29;
25;
5];
e = [-2;
-6;
15;
8];
df = [4;
9;
25;
16;
1];
ef = [-0.5;
-0.6666666666666666;
0.6;
0.5;
0];
b = [6, 10;
9, 4;
2, 9;
14, 65;
7, 23];
x = [2.5, 2;
2, -1;
1, -3;
-1, 6;
3, -5];
[xOut, ferr, berr, info] = nag_lapack_dptrfs(d, e, df, ef, b, x)
```
```

xOut =

2.5000    2.0000
2.0000   -1.0000
1.0000   -3.0000
-1.0000    6.0000
3.0000   -5.0000

ferr =

1.0e-13 *

0.2425
0.4663

berr =

0
0

info =

0

```
```function f07jh_example
d = [4;
10;
29;
25;
5];
e = [-2;
-6;
15;
8];
df = [4;
9;
25;
16;
1];
ef = [-0.5;
-0.6666666666666666;
0.6;
0.5;
0];
b = [6, 10;
9, 4;
2, 9;
14, 65;
7, 23];
x = [2.5, 2;
2, -1;
1, -3;
-1, 6;
3, -5];
[xOut, ferr, berr, info] = f07jh(d, e, df, ef, b, x)
```
```

xOut =

2.5000    2.0000
2.0000   -1.0000
1.0000   -3.0000
-1.0000    6.0000
3.0000   -5.0000

ferr =

1.0e-13 *

0.2425
0.4663

berr =

0
0

info =

0

```