Homework 6 Solutions - UCLA Mathematics
[Pages:3]Homework 6 Solutions
Igor Yanovsky (Math 151A TA)
Problem 1: Consider the numerical quadrature rule to approximate
1 0
f
(x)
dx
given
by
1
f (x) dx w1f (0) + w2f (x1).
0
Find the maximum possible degree of precision you can attain by appropriate choices of
w1, w2 and x1. With such choices of w1 and w2, approximate
1 0
x3
dx
and
compare
with
the exact value.
Solution: We want the formula
1
f (x) dx = w1f (0) + w2f (x1)
0
to hold for polynomials 1, x, x2, . . .. Plugging these into the formula, we obtain:
f (x) = x0 f (x) = x1 f (x) = x2
1
1 dx = x|10 = 1 = w1 ? 1 + w2 ? 1,
0
1
x2 1
1
x dx =
=
0
20 2
=
w1 ? 0 + w2 ? x1,
1
x2 dx
=
x3 1
=
1
0
30 3
=
w1 ? 0 + w2 ? x21.
We have 3 equations in 3 unknowns:
w1 + w2 = 1,
1
w2x1
=
, 2
w2x21
=
1 ,
3
or
w2 = 1 - w1,
1
x1(1 - w1)
=
, 2
x21(1 - w1)
=
1 .
3
Multiplying the second equation by x1 and subtracting the third equation, we obtain
x1
=
2 3
.
Then,
w2
=
3 4
and
w1
=
1 4
.
Thus, the quadrature formula is
1
1
32
f (x) dx = f (0) + f .
0
4
43
1
The accuracy of this quadrature formula is n = 2, since this formula holds for polynomials
1, x, x2.
We can check how well this formula approximates
1 0
x3
dx:
1
x3 dx
=
1
38
?0+ ?
2 = = 0.2222.
0
4
4 27 9
The exact value of this integral is
1
x3 dx
=
x4 1
1
= = 0.2500.
0
40 4
Problem 2: Determine constants a, b, c, d that will produce a quadrature formula
1
f (x) dx af (-1) + bf (1) + cf (-1) + df (1)
-1
that has degree of precision 3.
Solution: We want the formula
1
f (x) dx = af (-1) + bf (1) + cf (-1) + df (1)
-1
to hold for polynomials 1, x, x2, . . .. Plugging these into the formula, we obtain:
f (x) = x0 f (x) = x1 f (x) = x2 f (x) = x3
1
1 dx = x|1-1 = 2 = a ? 1 + b ? 1 + c ? 0 + d ? 0,
-1
1
x2 1
x dx =
= 0 = a ? (-1) + b ? 1 + c ? 1 + d ? 1,
-1
2 -1
1
x2 dx
=
x3 1
2 =
=
a ? 1 + b ? 1 + c ? (-2) + d ? 2,
-1
3 -1 3
1
x3 dx
=
x4 1
=0
=
a ? (-1) + b ? 1 + c ? 3 + d ? 3.
-1
4 -1
We have 4 equations in 4 unknowns:
a+b
= 2,
-a + b + c + d = 0, 2
a + b - 2c + 2d = , 3
-a + b + 3c + 3d = 0.
Solving this system, we obtain:
1
1
a = 1, b = 1, c = , d = - .
3
3
Thus, the quadrature formula with accuracy n = 3 is:
1
1
1
f (x) dx = f (-1) + f (1) + f (-1) - f (1).
-1
3
3
2
Computational Problem:
Approximate
2 0
x2
sin(-x)
dx
-2.4694834
by
the
follow-
ing quadrature rules to 10-6 accuracy and also find the size of h required for each rule.
Solution: See the code for the implementation of the composite numerical integration of the rules listed below.
The number of intervals specified below was sufficient to get an answer within 10-6 accuracy. The corresponding subinterval size is also specified.
(a) Composite left point rule. Number of intervals: n = 4, 000, 000; interval size: h = 5 ? 10-7.
(b) Composite right point rule. Number of intervals: n = 4, 000, 000; interval size: h = 5 ? 10-7.
(c) Composite midpoint rule. Number of intervals: n = 600; interval size: h = 0.0033.
(d) Composite trapezoidal rule. Number of intervals: n = 850; interval size: h = 0.0024.
(e) Composite Simpson's rule. Number of intervals: n = 18; interval size: h = 0.11.
3
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- multiplying by the conjugate university of washington
- the weierstrass approximation theorem college of engineering
- polynomials the legendre polynomial family department of scientific
- approximating functions by taylor polynomials clark science center
- math 115 hw 4 solutions colorado state university
- residue calculus part ii
- tionofrandomvariables university of illinois
- polynomial approximation of inverse sqrt function for fhe iacr
- math 314 lecture 12 14 2 limits and continuity brigham young university
- 11 the normal distribution and the central limit theorem 11 1 the
Related searches
- ucla health new patient forms
- grade 6 mathematics pdf
- class 11 mathematics solutions ncert
- ncert solutions class 6 science
- grade 6 mathematics textbook pdf
- ucla master of financial engineering
- practice and homework lesson 6 2
- grade 6 mathematics practice test
- grade 6 mathematics book
- mathematics grade 6 youtube
- ready mathematics grade 6 answers
- grade 6 mathematics worksheets