%Program 1.1 Regula False Method
%Computes approximate solution of f(x)=0
%Input: inline function f; a,b such that f(a)*f(b)<0, 
%     and iterartion number n positive integer
%Output: Bracketing interval [an,bn] and approximate  solution cn.
function [an,cn,bn] = RegFal1(f,a,b,n)
if sign(f(a))*sign(f(b)) >= 0
  error('f(a)f(b)<0 not satisfied!') %ceases execution
end
an=a; bn=b; 
fa=f(a);
fb=f(b);
c=b-fb*(b-a)/(fb-fa);
k = 1;
while k<n
  c=b-fb*(b-a)/(fb-fa);
  fc=f(c);
  if fc == 0              %c is a solution, done
    break
  end
  if sign(fc)*sign(fa)<0  %a and c make the new interval
    b=c;fb=fc; 
  else                    %c and b make the new interval
    a=c;fa=fc;
  end
  k=k+1;
end
an=a; fa=f(a);
bn=b; fb=f(b);
cn=b-fb*(b-a)/(fb-fa);
___________________________________________________________________________
 clear;
 format long
 f=inline('x*sin(x)-1','x');
A=zeros(10:3);
for j=1:10;
[x,y,u]=RegFal1(f,0,2,j);
A(j,1)=x; %left end point
A(j,2)=y; % intersection point 
A(j,3)=u; %right end point
end
  disp(' ')    %  a blank line
disp(' n         an                  cn                  bn ')
disp('-------------------------------------------------------------')
for k=1:10
  disp(sprintf('%2.0f    %7.15g    %7.15g   %7.15g',k,A(k,1),A(k,2),A(k,3)))
end
 
 n         an                  cn                  bn 
-------------------------------------------------------------
 1          0    1.09975017029462         2
 2    1.09975017029462    1.1212407359645         2
 3    1.09975017029462    1.11416119496263   1.1212407359645
 4    1.09975017029462    1.11415714303368   1.11416119496263
 5    1.09975017029462    1.11415714087308   1.11415714303368
 6    1.09975017029462    1.11415714087193   1.11415714087308
 7    1.09975017029462    1.11415714087193   1.11415714087193
 8    1.11415714087193    1.11415714087193   1.11415714087193
 9    1.11415714087193    1.11415714087193   1.11415714087193
10    1.11415714087193    1.11415714087193   1.11415714087193

%Comparison of methods

 clear;
 format long
 f=inline('x*sin(x)-1','x');
 iter=10;
A=zeros(iter:4);
for j=1:iter;
[x,y,u]=RegFal1(f,0,2,j);
[x1 y1 u1 w1]=bisect1(f,0,2,j);
A(j,1)=y; A(i,2)=f(y); 
A(j,3)=y1; A(i,4)=w1;
end
  disp(' ')    %  a blank line
disp(' n         cn                  f(cn)                  cn                 f(cn)')
disp('--------------------------------------------------------------------------------')
for k=1:iter
  disp(sprintf('%2.0f    %7.15g    %7.15g   %7.15g    %7.15g',k,A(k,1),A(k,2),A(k,3),A(k,4)))
end
n         cn                  f(cn)                  cn                 f(cn)
--------------------------------------------------------------------------------
 1    1.09975017029462    -0.0200192102426757         1    -0.158529015192104
 2    1.1212407359645    0.00983461086237658       1.5    0.496242479906082
 3    1.11416119496263    5.63035823186731e-006      1.25    0.186230774194483
 4    1.11415714303368    3.00226199456688e-009     1.125    0.0150510433614821
 5    1.11415714087308    1.60094160150948e-012    1.0625    -0.0718266313849869
 6    1.11415714087193    8.88178419700125e-016   1.09375    -0.0283617226631389
 7    1.11415714087193    -2.22044604925031e-016   1.109375    -0.00664277486029086
 8    1.11415714087193    2.22044604925031e-016   1.1171875    0.00420803401064274
 9    1.11415714087193    2.22044604925031e-016   1.11328125    -0.00121649041801597
10    1.11415714087193    2.22044604925031e-016   1.115234375    0.001496003657804

%Linear convergence of the Regula-Falsi Method 
%Solving f(x)=x^3-2x^2+1.5x=0 around x=0.
clear;
 format long
 f=inline('x^3-2*x^2+1.5*x','x');
A=zeros(20:3);
for j=1:20;
[x,y,u]=RegFal1(f,-1,1,j);
A(j,2)=abs(0-y);
if j>1
    A(j,3)=A(j,2)/A(j-1,2);
end
end
  disp(' ')    %  a blank line
disp(' n                 en              e(n)/e(n-1) ')
disp('-------------------------------------------------------')
for k=1:20
  disp(sprintf('%2.0f       %7.15g   %7.15g',k,A(k,2),A(k,3)))
end
 n               en              e(n)/e(n-1) 
-------------------------------------------------------
 1           0.8                     0
 2       0.642335766423358    0.802919708029197
 3       0.507240816253692   0.789681725304043
 4       0.390790151862501   0.770423316382038
 5       0.292974712875094   0.749698300939211
 6       0.213949069947435   0.730264628806549
 7       0.152685489795746   0.713653440200785
 8       0.106941252641754   0.700402197909009
 9       0.073828661942057   0.690366534132327
10       0.0504288271173981   0.683052161462389
11       0.0341840137058098   0.677866523173929
12       0.0230489505492406   0.674261096066763
13       0.0154840093511178   0.671788041630728
14       0.0103759488205704   0.670107372404895
15       0.0069412231515642   0.668972377523989
16       0.00463818878117337   0.668209144108579
17       0.00309690645233931   0.667697370342018
18       0.00206673559246874   0.667354866630728
19       0.00137877292652342   0.667125940806227
20       0.000919604398566524   0.666973060520784