conjugate_gradient2.hpp File Reference

Alternative conjugate gradient method for solving linear systems. More...

#include <cmath>
#include <stdexcept>
#include <string>

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Functions
template<typename Matrix , typename Vector >
Vector	conjugate_gradient2 (const Matrix &A, const Vector &b, Vector &x_vector, double tol=1e-5, int max_iter=1000)

Detailed Description

Alternative conjugate gradient method for solving linear systems.

This header provides an alternative implementation of the conjugate gradient method using operator overloading instead of method calls.

Function Documentation

conjugate_gradient2()

template<typename Matrix , typename Vector >

Vector conjugate_gradient2	(	const Matrix &	A,
		const Vector &	b,
		Vector &	x_vector,
		double	tol = 1e-5,
		int	max_iter = 1000
	)

Solves a system of linear equations \(Ax = b\) using the Conjugate Gradient method.

The update at each iteration \(k\):

\[ \alpha_k = \frac{r_k^T r_k}{d_k^T A d_k}, \qquad x_{k+1} = x_k + \alpha_k d_k \]

\[ r_{k+1} = r_k - \alpha_k A d_k, \qquad \beta_k = \frac{r_{k+1}^T r_{k+1}}{r_k^T r_k}, \qquad d_{k+1} = r_{k+1} + \beta_k d_k \]

Template Parameters

Matrix	The type of the matrix A.
Vector	The type of the vectors b and x_vector.

Parameters

A	The matrix A in the system of linear equations Ax = b.
b	The right-hand side vector b in the system of linear equations Ax = b.
x_vector	The initial guess for the solution vector. This vector will be updated in-place.
tol	The tolerance for the residual norm, used as the stopping criterion.
max_iter	The maximum number of iterations to perform.

Returns

The solution vector.

Exceptions

std::runtime_error

if the Conjugate Gradient method does not converge after the maximum number of iterations.