#Clear memory and load packages.
restart;
with(plots):
with(Statistics):

#Define vectors and variables.
actual:=<2,3.5136,5.8016,9.1136,13.7376,20>:
t:=<1,1.2,1.4,1.6,1.8,2>:
n:=6:
RMSE:=0:

#Perform a simple linear regression on the data.
P:=LinearFit([1,x],t,actual,x);

#Display the model errors.
printf("t    Actual P  Model P  Error\n");
for i from 1 to n do
   model:=subs(x=t[i],P):
   err:=actual[i]-model:
   RMSE:=RMSE+err^2:
   printf("%2d %7.4f% 10.4f% 9.4f\n",(i+(1/5)),actual[i],model,err);
end do:

#Display the root mean square error.
RMSE:=sqrt(RMSE/n);

#Plot the points and the model.
p1:=plot(t,actual,style=point,view=[0..7,8..26]):
p2:=plot(P,view=[0..7,8..26]):
display(p1,p2);

t    Actual P  Model P  Error

Error, (in fprintf) integer expected for integer format

#Clear memory and load package.
restart;  
with(LinearAlgebra):

#Initialise variables,matrices and vectors.
b:=<<18>,<-2>>:
c:=<<1>,<1>>:
i:=0:
P:=<1.08,1.37,1.56,1.61>;
t:=<5,10,15,20>;
tol:=1e-6:

#Initialise Gauss-Newton matrices.
n:=Dimension(t):
f:=Matrix(n,1):
J:=Matrix(n,2):

#Display initial parameter values.
printf("Gauss-Newton Method\n");
printf("-------------------\n");
printf("Before iterations,A = %f and B = %f\n",b(1),b(2));

#Perform the Gauss-Newton method.
while max(abs(evalf(c)))>tol do
  i:=i+1;
  for r from 1 to n do
     f(r,1):=evalf((In(b(1)+t[r])+b(2))-P[r]);
     J(r,1):=evalf(1/b(1)+t[r]);
     J(r,2):=evalf(1);
end do;
c:=Multiply(MatrixInverse(Multiply(Transpose(J),J)),Multiply(Transpose(J),f));
b:=b-c;
printf("After iterations %d, A = %f and B = %f\n",i,b(1),b(2));
end do:

Gauss-Newton Method
-------------------
Before iterations,A = 18.000000 and B = -2.000000
After iterations 1, A =

Error, (in fprintf) number expected for floating point format

#Clear memory.
restart;

#Initialise variables and arrays.
h:=0.1;
k:=Pi/30;
c:=1:
iter:=0:
tol:=1e-5;
w:=1.5;
u:=Array(0..10,0..10):
U:=Array(0..10,0..10):

#Calculate boundary u values.
for i from 0 to 10 do

   u[i,Pi/3]:=0;
   u[i,0]:=0;
end do:

for j from 0 to Pi/3 do
   u[0,j]:=0;   
   u[10,j]:=sin(3*j*k)/2;
end do:
#Perform the Gauss-Seidel method.
while c>tol do
   iter:=iter+1;
   if iter>99 then
      printf("\nGauss-Seidel method did not converge,change w.\n");
   break;
end if:
for i from 1 to 9 do
  for j from 0 to Pi/3 do
     U[i,j]:=u[i,j];
     u[i,j]:=(10*Pi^2*u[i-1,j]+9*u[i,j-1]+9*u[i,j+1]+2*Pi^2*u[i+1,j])/20;
     u[i,j]:=w*u[i,j]+(1-w)*U[i,j];
  end do:
end do:
c:=max(abs(u[1..9,1..9]-U[1..9,1..9]));
printf("After iteration %2d,the maximum change in u is %7.1e\n",iter,c);
end do:

#Display the final u values if method converged.
if iter<100 then
    printf("\nGauss-Seidel Method\n");
    printf("-------------------\n");
    for j from 10 by -1 to 0 do
      for i from 0 to 10 do
        printf("%6.4f",u[i,j]);
    end do:
    printf("\n");
end do:
end if:

Error, bad index into Array

Error, final value in for loop must be numeric or character

Error, final value in for loop must be numeric or character

Gauss-Seidel Method
-------------------
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000

#Clear memory.
restart;

#Initialise variables and arrays.
h:=0.1;
n:=10;
k:=0.1;  
m:=10;
t:=Array(0..m):
x:=Array(0..n):
u:=Array(0..n,0..m):

#Initialise the x array and the initial u(x,0) boundary.
for i from 0 to n do
    x[i]:=i*h;
    u[i,0]:=exp(x[i]);
end do:

#Initialise the t array and the u(x,t) side boundaries.
for j from 0 to m do
   t[j]:=j*k;
   u[0,j]:=0;
   u[n,j]:=t[j];
end do:

#Use the 2D CTCS explicit wave method.
for i from 1 to n-1 do
   u[i,1]:=(9*u[i-1,0]+14*u[i,0]+9*u[i+1,0])/32+[i]/10;
end do:
 
for j from 1 to m-1 do
  for i from 0 to n-1 do
   u[i,j+1]:=(9*u[i-1,j]+14**u[i,j]+9*u[i+1,j])/16-u[i,j-1];
  end do;
end do:

#Display the u(x,t)values.
printf("2D CTCS Explicit Wave Method\n");
printf("----------------------------\n");
printf("x\t\t t\t\t u\n");
for i from 0 to n do
   printf("% f\t% f\t% f\n",x[i],t[m],u[i,m]);
end do;

Error, Array index out of range

2D CTCS Explicit Wave Method
----------------------------
x                 t                 u
 0.000000         1.000000         0.000000
 0.100000         1.000000         0.000000
 0.200000         1.000000         0.000000
 0.300000         1.000000         0.000000
 0.400000         1.000000         0.000000
 0.500000         1.000000         0.000000
 0.600000         1.000000         0.000000
 0.700000         1.000000         0.000000
 0.800000         1.000000         0.000000
 0.900000         1.000000         0.000000
 1.000000         1.000000         1.000000