Prof. Muhammad Saeed 1.Nonlinear Equations 2.System of Linear Equations.
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Transcript of Prof. Muhammad Saeed 1.Nonlinear Equations 2.System of Linear Equations.
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Prof. Muhammad Saeed
1. Nonlinear Equations 2. System of Linear Equations
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2M.Sc. Physics
1.1.ErrErrors:ors: Personal Computer Number Constraints ( eps Etc. ) Truncation Round-Off Absolute (True ) Relative Approximate Relative Local Global Propagated
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M.Sc. Physics 3
2. Other Definitions Accuracy Precision
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M.Sc. Physics 4
3.3. Solution Of Solution Of Nonlinear Equations Nonlinear Equations (Roots ):(Roots ):
1. Bracketing Methods
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M.Sc. Physics 5
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0)(*)(
)()(
))((
0)(*)(
Linear Interpolation ( False Position )
Method
False Position Pitfalls
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M.Sc. Physics 6
2. Open MethodsFixed-Point Iteration
)(
0)(
xgx
xf
Fixed-Point Iteration
Convergence
Divergence
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M.Sc. Physics 7
Newton-Raphson
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)(
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Newton-Raphson Method
Newton-Raphson’s Pitfalls
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M.Sc. Physics 8
Secant
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M.Sc. Physics 9
4. Complex Roots Of PolynomialsMuller
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Muller Method
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M.Sc. Physics 10
Bairstow
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M.Sc. Physics 11
3. System Of Nonlinear Equations
Iterative Method
Newton’s Method
112112112
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12
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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
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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.
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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
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M.Sc. Physics 15