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

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

M.Sc. Physics3 2. Other Definitions Accuracy Precision

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

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

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

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

M.Sc. Physics8 Secant

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

M.Sc. Physics10 Bairstow

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

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

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.

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

M.Sc. Physics15