Solution Of Nolinear Equations 7-Numerical Analysis-Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Solution ofNon-Linear

Equations

Newton -Raphson

Method

Newton -Raphson

Method

An approximation tothe root is given by

0

1

0

0

(

)

(

)

f

x

x

x

f

x







Newton’s algorithmTo find a solution to

f(x)=

given an initialapproximation

p

0

INPUT

initial approximation

p

0

; tolerance TOL; maximum

number of iterations N

0

OUTPUT

approximate

solution p or message of failure

Step 3 Set p = p

0

- f ( p

0

) / f’ ( p

0

(compute p

i^

).

Step 4

If Abs (p – p

0

) < TOL

OUTPUT

( p );

(The procedure was successful.)

STOP

Step 5

Set

i = i + 1

Step 6

Set p

0

= p

(Update p

0

)

Step 7

OUTPUT

(The method failed after N

0

iterations, N

0

= ‘,N

0

)

The procedure was unsuccessful

STOP

SolutionThe Maple command will be asfollows,Fsolve ( cos (x) -x);



alg023();



This is Newton's Method Input the function F(x) in terms of xFor example:> cos(x)-xInput initial approximation> 0.7853981635Input tolerance> 0.

Newton's Method

I^

P

F(P)

1

-7.5487470e-

2

-7.5100000e-

3

0.0000000e-

Approximate solution = 0.73908513with F(P) = 0.0000000000Number of iterations = 3Tolerance = 5.0000000000e-

• Step
  • I =
    • Step
      • While i < N0 do Steps 3-
• Input maximum number ofiterations - no decimal point> 25Select output destination1. Screen2. Text fileEnter 1 or 2>
• Select amount of output1. Answer only2. All intermediateapproximationsEnter 1 or 2>
• Input maximum number ofiterations – no decimal point> 25Select output destination1. Screen2. Text fileEnter 1 or 2>
• Select amount of output1. Answer only2. All intermediateapproximationsEnter 1 or 2>