Numerical Methods MSA Libre

NUMERICAL SOLUTIONS:

Solved Examples

By

Mahmoud SAYED AHMED

Ph.D. Candidate

Department of Civil Engineering, Ryerson University

Toronto, Ontario 2013

Table of Contents Part I: Numerical Solution for Single Variable............................................................................................... 2

1.1. Newton-Raphson Method ............................................................................................................ 2

1.2. Secant Methods ............................................................................................................................ 4

Part Two: Numerical Solutions for Multiple Variables ................................................................................. 6

2.1. Generalized Newton-Raphson Method for Two Variables ........................................................... 6

2.2. Multi-dimensional case for Newton-Raphson Method ................................................................ 9

Appendix: Matrix ........................................................................................................................................ 10

Sayed-Ahmed, M. Ryerson University Jan. 2013

2

Part I: Numerical Solution for Single Variable

1.1. Newton-Raphson Method

The Newton-Raphson method (NRM) is powerful numerical method based on the simple idea of linear

approximation. NRM is usually home in on a root with devastating efficiency. It starts with initial guess,

where the NRM is usually very good if , and horrible if the guess are not close.

Question: Find the value of if using Newton-Raphson Method for three iterations? Answer: Start with guess value of The function equation should equal to zero; So the function equation; NRM:

The first iteration

then The absolute error, | | | |

The second iteration


The third iteration



3

Summary

Table 1: Newton-Raphson Method iteration results to three decimal places

Iteration Value of x Absolute error Exact Solution

1 2.741 9.45%

2 2.715 0.96% 2.714417617

3 2.714 0.009%

Figure 1: High initial solution

Important notes

At any time otherwise the process of iteration will not continue

0

10

20

30

40

50

60

70

2.72.82.933.13.23.33.4

Err

or,

%

Solution


4

1.2. Secant Methods

Question: Find the value of if using Secant Method for three iterations, where and ?

Answer: then

The first iteration, ( ) The absolute error

| | | | | | The second iteration, and

( ) The absolute error

| | | | | | The third iteration, and


| | | | | | The fourth iteration, and


| | | | | | The fifth iteration, and


| | | | | |


5

Summary

Table 2: Secant Method iteration results to three decimal places

Iteration Value of x Absolute error Exact Solution

1 3.353 63.92%

2 3.059 9.691% 2.714417617

3 2.906 5.26%

4 2.823 2.94%

5 2.7765 1.675%

Figure 2: High initial solution

0

10

20

30

40

50

60

70

2.72.82.933.13.23.33.4

Err

or,

%

Solution


6

Part Two: Numerical Solutions for Multiple Variables

2.1. Generalized Newton-Raphson Method for Two Variables

Question

For acceptable error less than 0.2, find the value of and

Solution

Where

[ ] Use Jacobian

[ ] [ ] The matrix notation

[ ]{ } { } [ ]{ } OR

{ } { } [ ] { }

[ ] [ ] [ ] Iteration;

The arbitrarily guess [ ] This scalar parameter which is adjusted to either less than 1 or more than 1 (=1 is the

original Newton Method) to force for convergence.

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Evaluate the function

[ ] [ ] [ [ ] ]


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[ ] [ ] [ = [ ] [ ] ] [ ] [ ] [ ]

Evaluate the function

Iteration; [ ] [ ] [ ] [ ]

Evaluate the function

( ) [ ] The iteration stops when results reached to specified tolerance of error | |

otherwise the process of iteration will continue.


8

Summary

Table 3: Generalized Newton-Raphson Method

Iteration Parameters Value Error, | |

0 1 1 100% 100%

1 2.1 1.3 52.38% 23.08%

2 1.8284 1.2122 14.85% 7.24%

Where the absolute error

| | | |

Figure 3: Graphical depiction of the solution of two simultaneous nonlinear equation

-6

-5

-4

-3

-2

-1

0

1

2

3

1 2 3

So

luti

on

Iteration

f(x1)

f(x2)


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2.2. Multi-dimensional case for Newton-Raphson Method

Talyor Series of m functions with n variables:

where

= J (Jacobian)

with m = n

Set

Advantages and Disadvantages:

The method is very expensive - It needs the function evaluation and then the derivative evaluation. If the tangent is parallel or nearly parallel to the x-axis, then the method does not converge. Usually Newton method is expected to converge only near the solution. The advantage of the method is its order of convergence is quadratic. Convergence rate is one of the fastest when it does converges

.

Source: (http://epoch.uwaterloo.ca/~ponnu/syde312/open_methods/page3.htm#example)


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Appendix: Matrix Inverse of Matrix

For 2 x 2 matrices

[ ] the inverse can be found using this formula [ ] [ ]

Multiplication of Matrix

for m x n matrix size and for n x p matrix size where for the order of m x p. Where is summed over all values of and the uses the Einsten summation

conventions.

Example: Square matrix and column vector

and The matrix product

( )

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Numerical Methods MSA Libre