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Newton's Interpolating Divided Difference Polynomial (MATLAB)

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0

$begingroup$

I'm trying to construct a polynomial in MATLAB using Newton's Interpolating Divided Difference Formula, and in doing so, generalize it to any size vector x and y. So far i was able to obtain the coefficients for the polynomial, but i'm unsure how to construct the polynomial itself. Here's what i've done so far:

%% Divided Difference Generalization
% Example:
 % Construct a Divided Difference Table for x0=2, x1=-1, x2=-6 and
 % f(x)=x^2+4x-5

 % x | 0th DD | 1'st DD | 2nd DD |
 % 2 | 7 | | |
 % -1 | -8 | -5 | |
 % -6 | 7 | -3 | 1 |


x=[2 -1 -6];
y=[7 -8 7];
size=length(x);
d=zeros(size,size);

xlength=length(x);
DDlim=length(x)-1;

D=zeros(xlength,xlength)
D(:,1)=y;

 for i=1:DDlim
 for j=1:i
 D(i+1,j+1)=(D(i+1,j)-D(i,j))/(x(i+1)-x(i-j+1));
 end
end

a=diag(D);

I'm trying to construct a general polynomial from the coefficents i've obtained from a to become:
begineqnarray
P_2(x) &&=&& 7+5(x-2)+1(x-2)(x+1)\
&&=&& x^2+4x-5
endeqnarray

And have it work for any number of points.

I'd appreciate any tips or help on to how to construct these polynomials without actually having to hardcode them. Thank you.

numerical-methods matlab

share|cite|improve this question

asked Mar 14 at 5:04

Mister_JMister_J

9615

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your first step is to study and code Horner's method for polynomials given in the standard basis. Afterwards you should extend Horner's method to polynomials given in Newton's form.
  $endgroup$
  – Carl Christian
  Mar 15 at 20:27
  

  
  
  
  

add a comment | 

0

$begingroup$

I'm trying to construct a polynomial in MATLAB using Newton's Interpolating Divided Difference Formula, and in doing so, generalize it to any size vector x and y. So far i was able to obtain the coefficients for the polynomial, but i'm unsure how to construct the polynomial itself. Here's what i've done so far:

%% Divided Difference Generalization
% Example:
 % Construct a Divided Difference Table for x0=2, x1=-1, x2=-6 and
 % f(x)=x^2+4x-5

 % x | 0th DD | 1'st DD | 2nd DD |
 % 2 | 7 | | |
 % -1 | -8 | -5 | |
 % -6 | 7 | -3 | 1 |


x=[2 -1 -6];
y=[7 -8 7];
size=length(x);
d=zeros(size,size);

xlength=length(x);
DDlim=length(x)-1;

D=zeros(xlength,xlength)
D(:,1)=y;

 for i=1:DDlim
 for j=1:i
 D(i+1,j+1)=(D(i+1,j)-D(i,j))/(x(i+1)-x(i-j+1));
 end
end

a=diag(D);

I'm trying to construct a general polynomial from the coefficents i've obtained from a to become:
begineqnarray
P_2(x) &&=&& 7+5(x-2)+1(x-2)(x+1)\
&&=&& x^2+4x-5
endeqnarray

And have it work for any number of points.

I'd appreciate any tips or help on to how to construct these polynomials without actually having to hardcode them. Thank you.

numerical-methods matlab

share|cite|improve this question

asked Mar 14 at 5:04

Mister_JMister_J

9615

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your first step is to study and code Horner's method for polynomials given in the standard basis. Afterwards you should extend Horner's method to polynomials given in Newton's form.
  $endgroup$
  – Carl Christian
  Mar 15 at 20:27
  

  
  
  
  

add a comment | 

$begingroup$

I'm trying to construct a polynomial in MATLAB using Newton's Interpolating Divided Difference Formula, and in doing so, generalize it to any size vector x and y. So far i was able to obtain the coefficients for the polynomial, but i'm unsure how to construct the polynomial itself. Here's what i've done so far:

%% Divided Difference Generalization
% Example:
 % Construct a Divided Difference Table for x0=2, x1=-1, x2=-6 and
 % f(x)=x^2+4x-5

 % x | 0th DD | 1'st DD | 2nd DD |
 % 2 | 7 | | |
 % -1 | -8 | -5 | |
 % -6 | 7 | -3 | 1 |


x=[2 -1 -6];
y=[7 -8 7];
size=length(x);
d=zeros(size,size);

xlength=length(x);
DDlim=length(x)-1;

D=zeros(xlength,xlength)
D(:,1)=y;

 for i=1:DDlim
 for j=1:i
 D(i+1,j+1)=(D(i+1,j)-D(i,j))/(x(i+1)-x(i-j+1));
 end
end

a=diag(D);

I'm trying to construct a general polynomial from the coefficents i've obtained from a to become:
begineqnarray
P_2(x) &&=&& 7+5(x-2)+1(x-2)(x+1)\
&&=&& x^2+4x-5
endeqnarray

And have it work for any number of points.

I'd appreciate any tips or help on to how to construct these polynomials without actually having to hardcode them. Thank you.

numerical-methods matlab

share|cite|improve this question

asked Mar 14 at 5:04

Mister_JMister_J

9615

$endgroup$

%% Divided Difference Generalization
% Example:
 % Construct a Divided Difference Table for x0=2, x1=-1, x2=-6 and
 % f(x)=x^2+4x-5

 % x | 0th DD | 1'st DD | 2nd DD |
 % 2 | 7 | | |
 % -1 | -8 | -5 | |
 % -6 | 7 | -3 | 1 |


x=[2 -1 -6];
y=[7 -8 7];
size=length(x);
d=zeros(size,size);

xlength=length(x);
DDlim=length(x)-1;

D=zeros(xlength,xlength)
D(:,1)=y;

 for i=1:DDlim
 for j=1:i
 D(i+1,j+1)=(D(i+1,j)-D(i,j))/(x(i+1)-x(i-j+1));
 end
end

a=diag(D);

share|cite|improve this question

asked Mar 14 at 5:04

Mister_JMister_J

9615

share|cite|improve this question

asked Mar 14 at 5:04

Mister_JMister_J

9615

asked Mar 14 at 5:04

Mister_JMister_J

9615

asked Mar 14 at 5:04

Mister_JMister_J

9615