Numerical Methods Solved Examples

In: Science

Submitted By photon88
Words 803
Pages 4
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

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:
Answer:

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

NRM:
The first iteration

( )
( )

then
( )
( )
( )

The absolute error,
|…...

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