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(Solved) Question 1 (a) Find an exact formula for the cubic polynomial


Question 1 (a) Find an exact formula for the cubic polynomial P3(x) =

x3 + ? ? ? such that


Math 128A, Spring 2016

 


 

Problem Set 07

 


 

Question 1 (a) Find an exact formula for the cubic polynomial P3 (x) =

 

x3 + ? ? ? such that

 

Z 1

 

P3 (x)q(x)dx = 0

 

1

 


 

for any quadratic polynomial q.

 

(b) Find exact formulas for the three roots x1 , x2 , x3 of the equation

 

P3 (x) = 0.

 

(c) Find exact formulas for the integration weights w1 , w2 , w3 such that

 

Z 1

 


 

1

 


 

q(x)dx =

 


 

3

 

X

 


 

wj q(xj )

 


 

j=1

 


 

exactly whenever q is a polynomial of degree 5.

 

(d) Given any real numbers a < b, ?nd exact formulas for points yj 2 [a, b]

 

and weights uj > 0 such that

 

Z b

 

a

 


 

q(x)dx =

 


 

3

 

X

 


 

uj q(yj )

 


 

j=1

 


 

whenever q is a polynomial of degree 5.

 

(e) Explain why each of the three factors in the error estimate

 

Z b

 

a

 


 

f (x)dx

 


 

3

 

X

 


 

uj f (yj ) = C6 f

 


 

(6)

 


 

(?)

 


 

j=1

 


 

Z b

 

a

 


 

(y

 


 

y1 )2 (y

 


 

y2 )2 (y

 


 

y3 )2 dy

 


 

is inevitable and determine the exact value of the constant C6 .

 

(f) Use your code from Question 2 to evaluate

 

E6 =

 


 

Z 1

 

0

 


 

(x

 


 

x1 )2 (x

 


 

x2 )2 (x

 


 

x3 )2 dx

 


 

to 3-digit accuracy.

 

Question 2 (a) Write, test and debug an adaptive 3-point Gaussian integration code gadap.m of the form

 

function [int, abt] = gadap(a, b, f, p, tol)

 

% a,b: interval endpoints with a < b

 

% f: function handle f(x, p) to integrate (p for user parameters)

 

% tol: User-provided tolerance for integral accuracy

 

% int: Approximation to the integral

 

% abt: Endpoints and approximations

 

1

 


 

Math 128A, Spring 2016

 


 

Problem Set 07

 


 

Build a list abt = {[a1 , b1 , t1 ], . . . , [an , bn , tn ]} of n intervals [aj , bj ] and apRb

 

proximate integrals tj ? ajj f (x)dx, computed with 3-point Gaussian integration. Initialize with n = 1 and [a1 , b1 ] = [a, b]. At each step j = 1, 2, . . .,

 

subdivide interval j into into left and right half-intervals l and r, and approximate the integrals tl and tr over each half-interval by 3-point Gaussian

 

quadrature. If

 

|tj

 


 

(tl + tr )| > tol max(|tj |, |tl | + |tr |)

 


 

add the half-intervals l and r and approximations tl and tr to the list. Otherwise, increment int by tj . Guard against in?nite loops and ?oating-point

 

issues as you see ?t and brie?y justify your design decisions in comments.

 

R

 

(b) Approximate the integral 01 x x dx using your code from (a). Measure

 

the total number of function evaluations required to obtain 12-digit accuracy.

 

Plot the accepted intervals. Compare your results with those obtained in the

 

previous problem set by Romberg integration.

 


 

2

 


 

 


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