Iterative Solution of Linear Equations - steepest descent method

　　1. Code

%% steepest descent method (symmetric positive definite for solving equations) 
%% linear equations M * X = b, M is a square matrix, X0 is the initial solution vector, epslion precision control 
function TSDM = The_steepest_descent_method (M, b , X0 , Epsilon) 
m = size (m); up = 1000; E = Floor (ABS (log (Epsilon))); 
X-(:,. 1) = X0; 
R & lt (:,. 1) = bM * X0; 
for K = . 1: up 
    Alpha = inner_product (R & lt (:, K), R & lt (:, K)) / inner_product (M * R & lt (:, K), R & lt (:, K)); 
    X-(:, K +. 1) = X-(:, K) + Alpha * R & lt (:, K); 
    R & lt (:, K +. 1) = bM * X-(:, K +. 1); 
    of X_DELTA in (:, K) = X-(:, K +. 1 ) the -X-(:, K); 
    IF sqrt (inner_product (of X_DELTA in (:, K), M * of X_DELTA in (:, K))) <Epsilon 
        BREAK; 
    End 
End 
DISP ( 'number of iterations:'); 
K. 1- 
TSDM = vpa (X (:, k), e); 
    the product %% 
    function IP = inner_product (M1, M2 ) 
        mAX = max (size (Ml)); 
        SUM = 0;
        for i = 1:MAX
            sum = sum+M1(i)*M2(i);
        end
        IP = sum;
    end
end

　　2. Examples

clear all
clc
for i = 1:4
    for j = 1:4
        if i == j
            M(i,j) = 2.1;
        else 
            M(i,j) = 1.5;
        end
    end
end
b = [1 2 3 4]';
X0 = [1 1 1 1]';
epsilon = 1e-4;

S = The_steepest_descent_method(M,b,X0,epsilon)

M\b

　　Results

The number of iterations is: 
ANS = 
    21 
S = 
  -2.12110743 
 -0.454511872 
   1.21208369 
   2.87867925 
ANS = 
   -2.1212 
   -0.4545 
    1.2121 
    2.8788 
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