Prof. Muhammad Saeed

1. Nonlinear Equations 2. System of Linear Equations

2M.Sc. Physics

1.1.ErrErrors:ors: Personal Computer Number Constraints ( eps Etc. ) Truncation Round-Off Absolute (True ) Relative Approximate Relative Local Global Propagated

M.Sc. Physics 3

2. Other Definitions Accuracy Precision

M.Sc. Physics 4

3.3. Solution Of Solution Of Nonlinear Equations Nonlinear Equations (Roots ):(Roots ):

1. Bracketing Methods

3213

213

21

0)(*)(2

0)(*)(

xxthenxfxif

xxx

xfxf

x2

x3

x1

M.Sc. Physics 5

ruur

ul

uluur

lu

xxthenxfxfif

xfxf

xxxfxx

xfxf

0)(*)(

)()(

))((

0)(*)(

Linear Interpolation ( False Position )

Method

False Position Pitfalls

M.Sc. Physics 6

2. Open MethodsFixed-Point Iteration

)(

0)(

xgx

xf

Fixed-Point Iteration

Convergence

Divergence

M.Sc. Physics 7

Newton-Raphson

10

0'

001 )(

)(

xx

xf

xfxx

Newton-Raphson Method

Newton-Raphson’s Pitfalls

M.Sc. Physics 8

Secant

3221

12

12223 )()()(

xxandxx

xfxf

xxxfxx

M.Sc. Physics 9

4. Complex Roots Of PolynomialsMuller

322110

223

21101

01

121010

121010

22

11

00

22

2

210

,,4

2

,,

)(),()(

,

)()(

)()(

)()(

)()()(

,,,0)(

xxxxxxacbb

cxx

xfcahbhh

a

xxfxfxf

xxhxxh

xfxf

xfxf

xfxf

cxxbxxaxf

xxxxf

p

p

p

p

Muller Method

M.Sc. Physics 10

Bairstow

1230

10

43210

2

432

23

14

0

1

1

:

11)(1

0)(

nnn

nn

ccca

bba

s

aaaaar

DivisionSyntheticEmploying

sandrthenxfoffactoraisxxif

axaxaxaxaxf

sssrrr

cc

cc

bc

bc

s

cc

cc

cb

cb

r

nn

nn

nn

nn

nn

nn

nn

nn

,,,

21

32

1

12

21

32

2

31

M.Sc. Physics 11

3. System Of Nonlinear Equations

Iterative Method

Newton’s Method

112112112

13

12

11

1

1

1

333

222

111

321

,,

)(

)(

)(

0),,(,0),,(,0),,(

zzzyyyxxx

xf

xf

xf

z

y

x

y

f

y

f

x

fz

f

y

f

x

fz

f

y

f

x

f

zyxfzyxfzyxf

M.Sc. Physics 12

4. Convergence Criteria

Fixed-Point Iteration Method:

Newton’s Method:

1)(,*)(1 iiii gege

1)(

)(*)(,*2/)( 2

21

xf

xfxfege ii

False Position Method:

1),(,*),( 111 iiiiii gege

Secant Method:

111 **2/),( iiiii eege

13M.Sc. Physics

4.4. About Solution of About Solution of Linear Equations:Linear Equations:

Pathologyi) Matrix is Singularii) System is ill-conditioned

( Small changes in input give rise to large changes

in the output) Pivoting and Scaling Norms of Matrices

i)

ii)

iii)

iv)

Condition No.

M.Sc. Physics 14

5.5. Solution of Solution of Linear Equations:Linear Equations:

Simple Iterative Method

Gauss-Seidel MethodThe diagonal element must be greater than the

off- diagonal element for each row to ensure the convergence.

Relaxation Method

M.Sc. Physics 15