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Fundamentals of Programmingfor Chemical Engineers-Introductory Concepts
Leandro T. De Castro
Methods in Solving
Engineering Problems
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Definition: Absolute Error and Relative Error
Let p* be an approximation to p
: =
: =
Round-off Errors
To minimize loss of accuracy due to round-offerrors:
a. Reformulate the problem
b. Reduce the number of error-producingcomputation
c. Use High-order digit arithmetic
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Convergence Rate
Suppose that a sequence {Xn} converges to a realnumber X*. We say the convergence is of order l (wherek>=1) provided that for some positive number C, thefollowing inequality will eventually true:
In general, higher values of k give faster convergence.
If a sequence converges at k=1, it is lineartly convergentIf a sequence converges at k=2, it convergesquadratically.
≤
Finding the root of f(x)=0
Methods:
Bisection Method
The Regula Falsi MethodThe Secant Method
Newton’s Method
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Bisection Method
- Based on the intermediate value theorem
Assumptions:
1. f(x) is continuous
2. On [a,b], f(a), f(b) have opposite signs
3. Tolerance = tol >0 is prescribed
Bisection Method
f(b)
f(a)
root r
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Bisection Method• Method will iteratively construct a sequence x
n thatconverges to the root r and will stop when it can beguaranteed that |r -xn| < tol .
• The difficulty though with the error expression, i.e., = is that the roor of a function is rarelyknown beforehand.
As the limiting criterion to the number of iterations, the
following may be used:1. < 2. <
3. < 4.
<
Algorithm for Bisection Method:
1. Choose an interval [a,b]
2. If f(a) and f(b) have opposite signs: start
3. Iterate: = + /24. If < : stop
5. Set a=x1 if f(x1) has same sign as f(a), else set b=x1
6. Goto 3
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Example: Consider the function
= 1
a. Show that f(x) has a root on the interval [0,1]
b. Use the bisection method to approximate this root with an absoluteerror <5e-3
c. How many iterations would it take the bisection method to guarantee
an absolute error < 1e-6
Regula Falsi
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Algorithm for Regula Falsi Method:
1. Choose an interval [xa, x b]
2. If f(xa) and f(xb) have opposite signs: start
3. Iterate: =
4. If error < tol; : stop
5. Set = if () has same sign as () , else set =
6. Goto 3
Example: Find the root of
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Seca t Method
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Algorithm for Secant Method:
1. Choose an interval [xn-1, x b]
2. If f(xn-1) and f(xn) have opposite signs: start
3. Iterate: =
4. If error < tol; : stop
5. Goto 3 using the two most recently generated points
Example: Find the root of
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Newton’s Method
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f(x) is a differentiable function that has a root
x= r which we wish to accurately approximate.
=
′
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