% POISSON EQUATION FINITE-DIFFERENCE ALGORITHM 12.1
%
% To approximate the solution to the Poisson equation
%            DEL(u) = F(x,y), a <= x <= b, c <= y <= d,
% SUBJECT TO BOUNDARY CONDITIONS:
%                 u(x,y) = G(x,y),
%     if x = a or x = b for c <= y <= d,
%     if y = c or y = d for a <= x <= b
%
% INPUT:   endpoints a, b, c, d; integers m, n; tolerance TOL;
%          maximum number of iterations M
%
% OUTPUT:  approximations W(I,J) to u(X(I),Y(J)) for each
%          I = 1,..., n-1 and J=1,..., m-1 or a message that the
%          maximum number of iterations was exceeded.
 syms('OK', 'A', 'B', 'C', 'D', 'N', 'M', 'TOL', 'NN');
 syms('M1', 'M2', 'N1', 'N2', 'H', 'K', 'I', 'X', 'J');
 syms('Y', 'W', 'V', 'VV', 'L', 'Z', 'E', 'LL', 'FLAG');
 syms('NAME', 'OUP', 's', 'x', 'y');
 TRUE = 1;
 FALSE = 0;
 fprintf(1,'This is the Finite-Difference Method for Elliptic Equations.\n');
 fprintf(1,'Input the functions F(X,Y) and G(X,Y) in terms of x and y \n');
 fprintf(1,'on separate lines. \n');
 fprintf(1,'For example:  x*exp(y) \n');
 fprintf(1,'              x*exp(y) \n');
 s = input(' ');
 F = inline(s,'x','y');
 s = input(' ');
 G = inline(s,'x','y');
 OK =FALSE;
 while OK == FALSE
 fprintf(1,'Input endpoints of interval (A,B) on X-axis\n');
 fprintf(1,'on separate lines.\n');
 A = input(' ');
 B = input(' ');
 fprintf(1,'Input endpoints of interval (C,D) on Y-axis\n');
 fprintf(1,'on separate lines.\n');
 C = input(' ');
 D = input(' ');
 if A >= B | C >= D
 fprintf(1,'Left endpoint must be less than right endpoint.\n');
 else
 OK = TRUE;
 end;
 end;
 OK = FALSE;
 while OK == FALSE
 fprintf(1,'Input number of intervals n on the X-axis and m\n');
 fprintf(1,'on the Y-axis on separate lines. \n');
 fprintf(1,'Note that both n and m should be larger than 2.\n');
 N = input(' ');
 M = input(' ');
 if M <= 2 | N <= 2
    fprintf(1,'Numbers must exceed 2.\n');
 else
 OK = TRUE;
 end;
 end;
 OK = FALSE;
 while OK == FALSE
 fprintf(1,'Input the Tolerance.\n');
 TOL = input(' ');
 if TOL <= 0
 fprintf(1,'Tolerance must be positive.\n');
 else
 OK = TRUE;
 end;
 end;
 OK = FALSE;
 while OK == FALSE
 fprintf(1,'Input the maximum number of iterations.\n');
 NN = input(' ');
 if NN <= 0
 fprintf(1,'Number must be a positive integer.\n');
 else
 OK = TRUE;
 end;
 end;
 fprintf(1,'Choice of output method:\n');
 fprintf(1,'1. Output to screen\n');
 fprintf(1,'2. Output to text file\n');
 fprintf(1,'Please enter 1 or 2.\n');
 FLAG = input(' ');
 if FLAG == 2
 fprintf(1,'Input the file name in the form - drive:\\name.ext\n');
 fprintf(1,'for example:  A:\\OUTPUT.DTA\n');
 NAME = input(' ','s');
 OUP = fopen(NAME,'wt');
 else
 OUP = 1;
 end;
 if OK == TRUE
 M1 = M-1;
 M2 = M-2;
 N1 = N-1;
 N2 = N-2;
% STEP 1
 H = (B-A)/N;
 K = (D-C)/M;
% STEPS 2 and 3 construct mesh points
% STEP 2
 X = zeros(1,N+1);
 Y = zeros(1,M+1);
 W = zeros(N+1,M+1);
 for I = 0 : N
 X(I+1) = A+I*H;
 end;
% STEP 3
 for J = 0 : M
 Y(J+1) = C+J*K;
 end;
% STEP 4
 for I = 1 : N1
 W(I+1,1) = G(X(I+1),Y(1));
 W(I+1,M+1) = G(X(I+1),Y(M+1));
 end;
 for J = 0 : M
 W(1,J+1) = G(X(1),Y(J+1));
 W(N+1,J+1) = G(X(N+1),Y(J+1));
 end;
 for I = 1 : N1
 for J = 1 : M1
 W(I+1,J+1) = 0;
 end;
 end;
% STEP 5
% use V for lambda, VV for mu
 V = H*H/(K*K);
 VV = 2*(1+V);
 L = 1;
 OK = FALSE;
% Z is a new value of W(I,J) to be used in computing the norm
% of the error E used in place of NORM
% STEP 6
 while L <= NN & OK == FALSE
% STEPS 7 through 20 perform Gauss-Seidel iterations
% STEP 7
 Z = (-H*H*F(X(2),Y(M1+1))+G(A,Y(M1+1))+V*G(X(2),D)+V*W(2,M2+1)+W(3,M1+1))/VV;
 E = abs( W(2,M1+1)-Z);
 W(2,M1+1) = Z;
% STEP 8
 for I = 2 : N2
 Z = (-H*H*F(X(I+1),Y(M1+1))+V*G(X(I+1),D)+W(I,M1+1)+W(I+2,M1+1)+V*W(I+1,M2+1))/VV;
 if abs(W(I+1,M1+1)-Z) > E
 E = abs( W(I+1,M1+1) - Z );
 end;
 W(I+1,M1+1) = Z;
 end;
% STEP 9
 Z = (-H*H*F(X(N1+1),Y(M1+1))+G(B,Y(M1+1))+V*G(X(N1+1),D)+W(N2+1,M1+1)+V*W(N1+1,M2+1))/VV;
 if abs( W(N1+1,M1+1)-Z) > E
 E = abs( W(N1+1,M1+1)-Z);
 end;
 W(N1+1,M1+1) = Z;
% STEP 10
 for LL = 2 : M2
 J = M2-LL+2;
% STEP 11
 Z = (-H*H*F(X(2),Y(J+1))+G(A,Y(J+1))+V*W(2,J+2)+V*W(2,J)+W(3,J+1))/VV;
 if abs(W(2,J+1)-Z) > E
 E = abs(W(2,J+1)-Z);
 end;
 W(2,J+1) = Z;
% STEP 12
 for I = 2 : N2
 Z = (-H*H*F(X(I+1),Y(J+1))+W(I,J+1)+V*W(I+1,J+2)+V*W(I+1,J)+W(I+2,J+1))/VV;
 if abs(W(I+1,J+1)-Z) > E
 E = abs(W(I+1,J+1)-Z);
 end;
 W(I+1,J+1) = Z;
 end;
% STEP 13
 Z = (-H*H*F(X(N1+1),Y(J+1))+G(B,Y(J+1))+W(N2+1,J+1)+V*W(N1+1,J+2)+V*W(N1+1,J))/VV;
 if abs(W(N1+1,J+1)-Z) > E
 E = abs(W(N1+1,J+1)-Z);
 end;
 W(N1+1,J+1) = Z;
 end;
% STEP 14
 Z = (-H*H*F(X(2),Y(2))+V*G(X(2),C)+G(A,Y(2))+V*W(2,3)+W(3,2))/VV;
 if abs(W(2,2)-Z) > E
 E = abs(W(2,2)-Z);
 end;
 W(2,2) = Z;
% STEP 15
 for I = 2 : N2
 Z = (-H*H*F(X(I+1),Y(2))+V*G(X(I+1),C)+W(I+2,2)+W(I,2)+V*W(I+1,3))/VV;
 if abs(W(I+1,2)-Z) > E
 E = abs(W(I+1,2)-Z);
 end;
 W(I+1,2) = Z;
 end;
% STEP 16
 Z = (-H*H*F(X(N1+1),Y(2))+V*G(X(N1+1),C)+G(B,Y(2))+W(N2+1,2)+V*W(N1+1,3))/VV;
 if abs(W(N1+1,2)-Z) > E
 E = abs(W(N1+1,2)-Z);
 end;
 W(N1+1,2) = Z;
% STEP 17
 if E <= TOL
% STEP 18
 fprintf(OUP, 'POISSON EQUATION FINITE-DIFFERENCE METHOD\n\n');
 fprintf(OUP,                                             '  I  J    X(I)        Y(J)         W(I,J)\n\n');
 for I = 1 : N1 
 for J = 1 : M1 
 fprintf(OUP, '%3d %2d %11.8f %11.8f %13.8f\n',I,J,X(I+1),Y(J+1),W(I+1,J+1));
 end;
 end;
 fprintf(OUP, 'Convergence occurred on iteration number: %d\n', L); 
% STEP 19
 OK = TRUE;
 else
% STEP 20
 L = L+1;
 end;
 end;
% STEP 21
 if L > NN 
 fprintf(1,'Method fails after iteration number %d\n', NN)
 end;
 if OUP ~= 1 
 fclose(OUP);
 fprintf(1,'Output file %s created successfully \n',NAME);
 end;
 end;