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Prof. Muhammad Saeed 1.Nonlinear Equations 2.System of Linear Equations

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2M.Sc. Physics 1.Erro rs: Personal Computer Number Constraints ( eps Etc. ) Truncation Round-Off Absolute (True ) Relative Approximate Relative Local Global Propagated

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M.Sc. Physics3 2. Other Definitions Accuracy Precision

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M.Sc. Physics4 3.Solution Of Nonlinear Equations (Roots ): Bisection Method 1. Bracketing Methods x2x2 x3x3 x1x1

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M.Sc. Physics5 Linear Interpolation ( False Position ) Method False Position Pitfalls

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M.Sc. Physics6 2. Open Methods Fixed-Point Iteration Convergence Divergence

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M.Sc. Physics7 Newton-Raphson Newton-Raphson Method Newton-Raphsons Pitfalls

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M.Sc. Physics8 Secant

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M.Sc. Physics9 4. Complex Roots Of Polynomials Muller Muller Method

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M.Sc. Physics10 Bairstow

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M.Sc. Physics11 3.System Of Nonlinear Equations Iterative Method Newtons Method

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M.Sc. Physics12 4.Convergence Criteria Fixed-Point Iteration Method: Newtons Method: False Position Method: Secant Method:

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13M.Sc. Physics 4.About Solution of Linear Equations: Pathology i)Matrix is Singular ii)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. Physics14 5.Solution of Linear Equations: Simple Iterative Method Gauss-Seidel Method The 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. Physics15

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