2 ObjectiveFinding 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- SecantSolving Ax = b or Ax – b = 0, or finding multiple x's that simultaneously satisfy a system of linear equations- Gauss-Seidel- JacobiVariation 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?
6 Fixed Point Iteration – Converging Criteria For solving f(x) = 0 (one equation, one unknown), we have the updating formulaTrough analysis, we derived the following relationshipwhich tells us convergence is guaranteed if
7 Fixed Point Iteration – Converging Criteria For solving two equations with two unknowns, we have the updating formulaThrough 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 haveIdea: 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 haveSingle-equation form
10 Newton-Raphson (two equations, two unknowns) Want to find x and y that satisfyFrom 1st-Order Taylor Series Approximation, we haveUsing 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 yieldscontinue …
12 Newton-Raphson (two equations, two unknowns) Solving the equations algebraically yieldsAlternatively, we may solve for xi+1 and yi+1 using well-known methods for solving systems of linear equations
13 Newton-Raphson Example To solveFirst evaluateWith x0 = 1.5, y0 = 3.5, we havecontinue …
14 Newton-Raphson Example From these two formula, we can then calculate x1 and y1 asThese 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 satisfyFrom 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 yieldsThese equations can be expressed in matrix form aswhere
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.