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Systems of Non-Linear Equations

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Presentation on theme: "Systems of Non-Linear Equations"— Presentation transcript:

1 Systems of Non-Linear Equations

2 Objective Finding the roots of a set of simultaneous nonlinear equations (n equations, n unknowns). where each of fi(x1, … xn) cannot be expressed in the form

3 Review of iterative methods for finding unknowns
Finding x that satisfies f(x) = 0 (one equation, one unknown) - Fixed-Point Iteration - Newton-Raphson - Secant Solving Ax = b or Ax – b = 0, or finding multiple x's that simultaneously satisfy a system of linear equations - Gauss-Seidel - Jacobi Variation of Fixed-Point Iteration

4 Fixed Point Iteration To solve We can create updating formula as or
Which formula will converge? What initial points should we pick?

5 Fixed Point Iteration Diverging Converging

6 Fixed Point Iteration – Converging Criteria
For solving f(x) = 0 (one equation, one unknown), we have the updating formula Trough analysis, we derived the following relationship which tells us convergence is guaranteed if

7 Fixed Point Iteration – Converging Criteria
For solving two equations with two unknowns, we have the updating formula Through similar reasoning, we can demonstrate that convergence can be guaranteed if

8 Fixed-Point Iteration – Summary
Updating formula is easy to construct, but updating formula that satisfy (guarantees convergence) is not easy to construct. Slow convergent rate

9 Newton-Raphson (one equation, one unknown)
Want to find the root of f(x) = 0. From 1st-Order Taylor Series Approximation, we have Idea: use the slope at xi to predict the location of the root. If xi+1 is the root, then f(xi+1) = 0. Thus we have Single-equation form

10 Newton-Raphson (two equations, two unknowns)
Want to find x and y that satisfy From 1st-Order Taylor Series Approximation, we have Using similar reasoning, we have ui+1 = 0 and vi+1 = 0. continue …

11 Newton-Raphson (two equations, two unknowns)
Replacing ui+1 = 0 and vi+1 = 0 in the equations yields continue …

12 Newton-Raphson (two equations, two unknowns)
Solving the equations algebraically yields Alternatively, we may solve for xi+1 and yi+1 using well-known methods for solving systems of linear equations

13 Newton-Raphson Example
To solve First evaluate With x0 = 1.5, y0 = 3.5, we have continue …

14 Newton-Raphson Example
From these two formula, we can then calculate x1 and y1 as These process can be repeated until a "good enough" approximation is obtained.

15 Newton-Raphson (n equations, n unknowns)
Want to find xi (i = 1, 2, …, n) that satisfy From 1st-Order Taylor Series Approximation, we have

16 Newton-Raphson (n equations, n unknowns)
For each k = 0, 1, 2, …, n, setting fk,i+1 = 0 yields These equations can be expressed in matrix form as where

17 Newton-Raphson – Summary
Updating formula is not convenient to construct. Excellent initial guesses are usually required to ensure convergence. If the iteration converges, it converges quickly.

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