Conjugate gradient

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A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate gradient (in red) for minimizing the quadratic form associated with a given linear system. Conjugate gradient, assuming exact arithmetics, converges in at most n steps where n is the size of the matrix of the system (here n=2).

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems which are too large to be handled by direct methods such as the Cholesky decomposition. Such systems arise regularly when numerically solving partial differential equations.

The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization.

The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear equations.

1 Description of the method
2 Conjugate gradient on the normal equations
3 See also
4 References
5 External links

Description of the method

Suppose we want to solve the following system of linear equations

Ax = b

where the n-by-n matrix A is symmetric (i.e., A^T = A), positive definite (i.e., x^TAx > 0 for all non-zero vectors x in Rⁿ), and real.

We denote the unique solution of this system by x_*.

The conjugate gradient method as a direct method

We say that two non-zero vectors u and v are conjugate (with respect to A) if

$\mathbf{u}^{\mathrm{T}} \mathbf{A} \mathbf{v} = \mathbf{0}.$

Since A is symmetric and positive definite, the left-hand side defines an inner product

$\langle \mathbf{u},\mathbf{v} \rangle_\mathbf{A} := \langle \mathbf{A}^{\mathrm{T}} \mathbf{u}, \mathbf{v}\rangle = \langle \mathbf{A} \mathbf{u}, \mathbf{v}\rangle = \langle \mathbf{u}, \mathbf{A}\mathbf{v} \rangle = \mathbf{u}^{\mathrm{T}} \mathbf{A} \mathbf{v}.$

So, two vectors are conjugate if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation: if u is conjugate to v, then v is conjugate to u. (Note: This notion of conjugate is not related to the notion of complex conjugate.)

Suppose that {p_k} is a sequence of n mutually conjugate directions. Then the p_k form a basis of Rⁿ, so we can expand the solution x_* of Ax = b in this basis:

$\mathbf{x}_* = \sum^{n}_{i=1} \alpha_i \mathbf{p}_i$

The coefficients are given by

$\mathbf{A}\mathbf{x}_* = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.$

$\mathbf{p}_k^{\mathrm{T}} \mathbf{A}\mathbf{x}_* = \sum^{n}_{i=1} \alpha_i\mathbf{p}_k^{\mathrm{T}} \mathbf{A} \mathbf{p}_i= \mathbf{p}_k^{\mathrm{T}} \mathbf{b}.$

$\alpha_k = \frac{\mathbf{p}_k^{\mathrm{T}} \mathbf{b}}{\mathbf{p}_k^{\mathrm{T}} \mathbf{A} \mathbf{p}_k} = \frac{\langle \mathbf{p}_k, \mathbf{b}\rangle}{\,\,\,\langle \mathbf{p}_k, \mathbf{p}_k\rangle_\mathbf{A}} = \frac{\langle \mathbf{p}_k, \mathbf{b}\rangle}{\,\,\,\|\mathbf{p}_k\|_\mathbf{A}^2}.$

This result is perhaps most transparent by considering the inner product defined above.

This gives the following method for solving the equation Ax = b. We first find a sequence of n conjugate directions and then we compute the coefficients α_k.

The conjugate gradient method as an iterative method

If we choose the conjugate vectors p_k carefully, then we may not need all of them to obtain a good approximation to the solution x_*. So, we want to regard the conjugate gradient method as an iterative method. This also allows us to solve systems where n is so large that the direct method would take too much time.

We denote the initial guess for x_* by x₀. We can assume without loss of generality that x₀ = 0 (otherwise, consider the system Az = b − Ax₀ instead). Note that the solution x_* is also the unique minimizer of the quadratic form

$f(\mathbf{x}) = \frac12 \mathbf{x}^{\mathrm{T}} \mathbf{A}\mathbf{x} - \mathbf{b}^{\mathrm{T}} \mathbf{x} , \quad \mathbf{x}\in\mathbf{R}^n.$

This suggests taking the first basis vector p₁ to be the gradient of f at x = x₀, which equals Ax₀−b or, since x₀ = 0, −b. The other vectors in the basis will be conjugate to the gradient, hence the name conjugate gradient method.

Let r_k be the residual at the kth step:

$\mathbf{r}_k = \mathbf{b} - \mathbf{Ax}_k. \,$

Note that r_k is the negative gradient of f at x = x_k, so the gradient descent method would be to move in the direction r_k. Here, we insist that the directions p_k are conjugate to each other, so we take the direction closest to the gradient r_k under the conjugacy constraint. This gives the following expression:

$\mathbf{p}_{k+1} = \mathbf{r}_k - \frac{\mathbf{p}_k^{\mathrm{T}} \mathbf{A} \mathbf{r}_k}{\mathbf{p}_k^{\mathrm{T}}\mathbf{A} \mathbf{p}_k} \mathbf{p}_k$

(see the picture at the top of the article for the effect of the conjugacy constraint on convergence).

The resulting algorithm

After some simplifications, this results in the following algorithm for solving Ax = b where A is a real, symmetric, positive-definite matrix. The input vector x₀ can be an approximate initial solution or 0.

$r_0 := b - A x_0 \,$

$p_0 := r_0 \,$

$k := 0 \,$

repeat

$\alpha_k := \frac{r_k^\top r_k}{p_k^\top A p_k} \,$

$x_{k+1} := x_k + \alpha_k p_k \,$

$r_{k+1} := r_k - \alpha_k A p_k \,$

if r_k+1 is "sufficiently small" then exit loop end if

$\beta_k := \frac{r_{k+1}^\top r_{k+1}}{r_k^\top r_k} \,$

$p_{k+1} := r_{k+1} + \beta_k p_k \,$

$k := k + 1 \,$

end repeat

The result is $x_{k+1} \,$

Example of conjugate gradient method for Octave

 function [x] = conjgrad(A,b,x0)
  
    r = b - A*x0;
    w = -r;
    z = A*w;
    a = (r'*w)/(w'*z);
    x = x0 + a*w;
    B = 0;
  
    for i = 1:size(A)(1);
       r = r - a*z;
       if( norm(r) < 1e-10 )
            break;
       endif
       B = (r'*z)/(w'*z);
       w = -r + B*w;
       z = A*w;
       a = (r'*w)/(w'*z);
       x = x + a*w;
    end
  
 end

Preconditioner

A preconditioner is a matrix P such that P^-1A has a smaller condition number (κ) than A and so solving P^-1Ax=P^-1b is faster than solving Ax=b (see preconditioned conjugate gradient method).

Conjugate gradient on the normal equations

The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A^TA and right-hand side vector A^Tb, since A^TA is a symmetric positive (semi-)definite matrix for any A. The result is conjugate gradient on the normal equations (CGNR).

A^TAx = A^Tb

As an iterative method, it is not necessary to form A^TA explicitly in memory but only to perform the matrix-vector and transpose matrix-vector multiplications. Therefore CGNR is particularly useful when A is a sparse matrix since these operations are usually extremely efficient. However the downside of forming the normal equations is that the condition number κ(A^TA) is equal to κA² and so the rate of convergence of CGNR may be slow. Finding a good preconditioner is often an important part of using the CGNR method.

Several algorithms have been proposed (e.g., CGLS, LSQR). The LSQR algorithm purportedly has the best numerical stability when A is ill-conditioned, i.e., A has a large condition number.

References

The conjugate gradient method was originally proposed in

Hestenes, Magnus R.; Stiefel, Eduard (December, 1952). "Methods of Conjugate Gradients for Solving Linear Systems". Journal of Research of the National Bureau of Standards 49 (6), http://nvl.nist.gov/pub/nistpubs/jres/049/6/V49.N06.A08.pdf.

Descriptions of the method can be found in the following text books:

Kendell A. Atkinson (1988), An introduction to numerical analysis (2nd ed.), Section 8.9, John Wiley and Sons. ISBN 0-471-50023-2.
Mordecai Avriel (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0.
Gene H. Golub and Charles F. Van Loan, Matrix computations (3rd ed.), Chapter 10, Johns Hopkins University Press. ISBN 0-8018-5414-8.