Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 4.3, Problem 3CP
To determine
To find out the Hilbert matrix using matlab code for given condition in the problem.
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Consider the differential equation y' = 5y with initial condition y(0) :
The actual solution is y(1) = 207.78 help (numbers)
= 1.4.
We wish to analyze what happens to the error when estimating y(1) via Euler's method.
Start with step size h = 1 (1 step). Compute
y(1)
Error
8.4 help (numbers)
199.38 help (numbers)
Note: Remember that the error is the absolute value!
Let us half the step size to h
=
0.5 (2 steps). Compute
y(1) ≈ 17.15 help (numbers)
Error =
190.63 help (numbers)
The error went down by the factor:
Error
Previous error
Let us half the step size to h = 0.25 (4 steps). Compute
y(1) 35.88046875 help (numbers)
Error = 171.90 help (numbers)
help (numbers)
The error went down by the factor:
Error
Previous error
help (numbers)
Euler's method is a first order method so we expect the error to go down by a factor of 0.5
each halving. Of course, that's only very approximate, so the numbers you get above are not
exactly 0.5.
Book: Section 1.7 of Notes on Diffy Qs
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Chapter 4 Solutions
Numerical Analysis
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