fundamentals of programming for chemical engineers methods in solving engg problems

7/26/2019 Fundamentals of Programming for Chemical Engineers Methods in Solving Engg Problems

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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.

=

′

fundamentals of programming for chemical engineers methods in solving engg problems

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