# Gradient Methods

Linear system AX = B is solved by finding a minimum of the functional F(X) = 1/2 X^{T}A X - X

^{T}B .

**Assumptions on the matrix A **

- A is symmetric: A = A^{T}

- A is positive definite, so for example the leading minors could be checked,
or eigenvalues of A, or some other sufficient condition on A

**Method of Gradient Descent (or Steepest Descent) **

>> A = [ 4 1 1 0; 1 4 1 1; 1 1 4 1; 0 1 1 4] % the matrix >> B = [ -2; 4; 10; 14] % the right hand side >> X = [ 0; 0; 0; 0] % the first approximation - any vector

repeat the following 4 steps, until the norm of the residual is less than some chosen tolerance:

>> R = B - A * X % residual - the direction for searching the new approx. >> norm(R) % norm of the residual - the criterion of convergence >> alfa = (R' * R) / (R' * (A * R)) % the coefficient (assuming norm(R) > 0) >> X = X + alfa * R % the new approximationAfter 10 iterations the result X = [ -1; 0; 2; 3] is obtained with precision of 4 decimal places, euclides norm of the residual is less than 0.001.

More effective programming - saves one multiplication by the matrix when computing the residual:

(in this case we must choose the first approx. equal to zero vector, however)

>> X = [ 0; 0; 0; 0] >> R = B % only holds for X = zero vector

repeat the following steps, until the norm of the residual is less than some chosen tolerance:

>> Q = A * R ; >> alfa = (R' * R) / (R' * Q) % the coefficient (assuming norm(R) > 0) >> X = X + alfa * R % the new approximation >> R = R - alfa * Q % residual - the direction for searching the new approx. >> norm(R) % norm of the residual - the criterion of convergence

The function using the Method of Gradient Descent (as a file nejv_spad.m):

function [X, res, i] = nejv_spad( A, B, tol, numit) % [X, res, i] = nejv_spad( A, B, tol, numit) % input: A - the system matrix (symmetric positive definite, it is not checked) % B - the right hand side % tol - tolerance for norm of the residual % numit - maximal number of iterations % output: X the solution of AX=B computed with given tolerance or after numit iterations % res - norm of the residual % i - number of iterations X = zeros(size(B)); % the first approx. (zero vector of size of B) R = B; % residual res = norm(R); % norm of the residual i = 0; % iterations while ((i < numit) && (res > tol)) i = i+1; Q = A * R; alfa = (R' * R) / (R' * Q); X = X + alfa * R; R = R - alfa * Q; res = norm(R); end % while end % function

function can be called in different ways (assume A and B are defined already):

>> [X, r, i] = nejv_spad(A, B, 0.1 ,6); % all the results saved in X, r, i >> X = nejv_spad(A, B, 0.001, 15); % X only >> nejv_spad(A, B, 0.01, 10) % results will be displayed on the screen >> help nejv_spad % the help - displays the leading comments

**Conjugate Gradients Method**

>> X = [ 0; 0; 0; 0] % the first approx. - any vector >> R = B - A*X >> V = R

repeat the following steps, until the norm of the residual is less than some chosen tolerance:

>> Q = A * V ; >> alfa = (V' * R) / (V' * Q) >> X = X + alfa * V % the new approximation >> R = R - alfa * Q % the residual >> norm(R) % norm of the residual - the criterion of convergence >> beta = - (V' * A * R) / (V' * Q) >> V = R + beta * V % the new direction for searching the minimum