NAG FL Interfacef07jbf (dptsvx)

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1Purpose

f07jbf uses the factorization
 $A=LDLT$
to compute the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ symmetric positive definite tridiagonal matrix and $X$ and $B$ are $n×r$ matrices. Error bounds on the solution and a condition estimate are also provided.

2Specification

Fortran Interface
 Subroutine f07jbf ( fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, nrhs, ldb, ldx Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: d(*), e(*), b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: df(*), ef(*), x(ldx,*) Real (Kind=nag_wp), Intent (Out) :: rcond, ferr(nrhs), berr(nrhs), work(2*n) Character (1), Intent (In) :: fact
#include <nag.h>
 void f07jbf_ (const char *fact, const Integer *n, const Integer *nrhs, const double d[], const double e[], double df[], double ef[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double *rcond, double ferr[], double berr[], double work[], Integer *info, const Charlen length_fact)
The routine may be called by the names f07jbf, nagf_lapacklin_dptsvx or its LAPACK name dptsvx.

3Description

f07jbf performs the following steps:
1. 1.If ${\mathbf{fact}}=\text{'N'}$, the matrix $A$ is factorized as $A=LD{L}^{\mathrm{T}}$, where $L$ is a unit lower bidiagonal matrix and $D$ is diagonal. The factorization can also be regarded as having the form $A={U}^{\mathrm{T}}DU$.
2. 2.If the leading $i×i$ principal minor is not positive definite, then the routine 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, ${\mathbf{info}}=\mathbf{n}+{\mathbf{1}}$ is returned as a warning, but the routine still goes on to solve for $X$ and compute error bounds as described below.
3. 3.The system of equations is solved for $X$ using the factored form of $A$.
4. 4.Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

4References

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 https://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

5Arguments

1: $\mathbf{fact}$Character(1) Input
On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.
${\mathbf{fact}}=\text{'F'}$
df and ef contain the factorized form of the matrix $A$. df and ef will not be modified.
${\mathbf{fact}}=\text{'N'}$
The matrix $A$ will be copied to df and ef and factorized.
Constraint: ${\mathbf{fact}}=\text{'F'}$ or $\text{'N'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{d}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ diagonal elements of the tridiagonal matrix $A$.
5: $\mathbf{e}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
6: $\mathbf{df}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array df must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, df must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\text{'N'}$, df contains the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
7: $\mathbf{ef}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array ef must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, ef must contain the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
On exit: if ${\mathbf{fact}}=\text{'N'}$, ef contains the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
8: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ right-hand side matrix $B$.
9: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07jbf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{x}\left({\mathbf{ldx}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n×r$ solution matrix $X$.
11: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07jbf is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12: $\mathbf{rcond}$Real (Kind=nag_wp) Output
On exit: the reciprocal condition number of the matrix $A$. If rcond is less than the machine precision (in particular, if ${\mathbf{rcond}}=0.0$), the matrix is singular to working precision. This condition is indicated by a return code of ${\mathbf{info}}=\mathbf{n}+{\mathbf{1}}$.
13: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$Real (Kind=nag_wp) array Output
On exit: the forward error bound for each solution vector ${\stackrel{^}{x}}_{j}$ (the $j$th column of the solution matrix $X$). If ${x}_{j}$ is the true solution corresponding to ${\stackrel{^}{x}}_{j}$, ${\mathbf{ferr}}\left(j\right)$ is an estimated upper bound for the magnitude of the largest element in (${\stackrel{^}{x}}_{j}-{x}_{j}$) divided by the magnitude of the largest element in ${\stackrel{^}{x}}_{j}$.
14: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$Real (Kind=nag_wp) array Output
On exit: the component-wise relative backward error of each 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).
15: $\mathbf{work}\left(2×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
16: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and 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 $⟨\mathit{\text{value}}⟩$ 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.
${\mathbf{info}}={\mathbf{n}}+1$
$D$ 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.

7Accuracy

For each right-hand side vector $b$, the computed solution $\stackrel{^}{x}$ is the exact solution of a perturbed system of equations $\left(A+E\right)\stackrel{^}{x}=b$, where
 $|E| ≤ c (n) ε |R| |RT| , where ​ R = L D12 ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 10.1 of Higham (2002) for further details.
If $x$ is the true solution, then the computed solution $\stackrel{^}{x}$ satisfies a forward error bound of the form
 $‖x-x^‖∞ ‖x^‖∞ ≤ wc cond(A,x^,b)$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖|{A}^{-1}|\left(|A||\stackrel{^}{x}|+|b|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$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.

8Parallelism and Performance

f07jbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07jbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The number of floating-point operations required for the factorization, and for the estimation of the condition number of $A$ is proportional to $n$. The number of floating-point operations required for the solution of the equations, and for the estimation of the forward and backward error is proportional to $nr$, where $r$ is the number of right-hand sides.
The condition estimation is based upon Equation (15.11) of Higham (2002). For further details of the error estimation, see Section 4.4 of Anderson et al. (1999).
The complex analogue of this routine is f07jpf.

10Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = ( 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 )$
and
 $B = ( 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 ) .$
Error estimates for the solutions and an estimate of the reciprocal of the condition number of $A$ are also output.

10.1Program Text

Program Text (f07jbfe.f90)

10.2Program Data

Program Data (f07jbfe.d)

10.3Program Results

Program Results (f07jbfe.r)