conjugate_gradient.hpp File Reference

Conjugate gradient method for solving linear systems. More...

#include <cmath>
#include <stdexcept>
#include <string>
#include <utility>
#include <vector>

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Classes
class	Vector0
	A simple vector class for conjugate gradient calculations. More...
class	Matrix0
	A simple matrix class for conjugate gradient calculations. More...

Functions
Vector0	operator+ (Vector0 lhs, const Vector0 &rhs)
	Element-wise vector addition.
Vector0	operator- (Vector0 lhs, const Vector0 &rhs)
	Element-wise vector subtraction.
Vector0	operator* (Vector0 v, double scalar)
	Scalar-vector multiplication (vector * scalar)
Vector0	operator* (double scalar, Vector0 v)
	Scalar-vector multiplication (scalar * vector)
template<typename Matrix0 , typename Vector0 >
Vector0	conjugate_gradient (const Matrix0 &A, const Vector0 &b, const Vector0 *x0=nullptr, double tol=1e-5, int max_iter=1000)
	Solve Ax = b using the conjugate gradient method.

Detailed Description

Conjugate gradient method for solving linear systems.

This header provides classes and functions for solving linear systems using the conjugate gradient method. It includes a simple Vector and Matrix class for demonstration purposes.

Function Documentation

conjugate_gradient()

template<typename Matrix0 , typename Vector0 >

Vector0 conjugate_gradient ( const Matrix0 &  A, const Vector0 &  b, const Vector0 *  x0 = nullptr, double  tol = 1e-5, int  max_iter = 1000  )

inline

Solve Ax = b using the conjugate gradient method.

The Conjugate Gradient method iteratively solves \(Ax = b\) where \(A\) is symmetric positive-definite.

The update at each iteration \(k\) is:

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

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

Template Parameters

Matrix0	The matrix type, which must support matrix-vector multiplication.
Vector0	The vector type, which must support vector operations.

Parameters

A	The matrix A in the linear system Ax = b.
b	The right-hand side vector b in the linear system Ax = b.
x0	An optional initial guess for the solution vector.
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.

operator*() [1/2]

Vector0 operator* ( double  scalar, Vector0  v  )

inline

Scalar-vector multiplication (scalar * vector)

Parameters

[in]	scalar	The scalar value
[in]	v	The vector to multiply

Returns

The resulting vector

operator*() [2/2]

Vector0 operator* ( Vector0  v, double  scalar  )

inline

Scalar-vector multiplication (vector * scalar)

Parameters

[in]	v	The vector to multiply
[in]	scalar	The scalar value

Returns

The resulting vector

operator+()

Vector0 operator+ ( Vector0  lhs, const Vector0 &  rhs  )

inline

Element-wise vector addition.

Parameters

[in]	lhs	Left-hand side vector
[in]	rhs	Right-hand side vector

Returns

The resulting vector

operator-()

Vector0 operator- ( Vector0  lhs, const Vector0 &  rhs  )

inline

Element-wise vector subtraction.

Parameters

[in]	lhs	Left-hand side vector
[in]	rhs	Right-hand side vector

Returns

The resulting vector