Systems of NonLinear Equations

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Systems of Non-Linear Equations
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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
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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
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Fixed Point Iteration

• To solve

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

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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 xi1 is the root, then f(xi1)
0. Thus we have
Single-equation form
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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 ui1 0 and
vi1 0.
continue
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Newton-Raphson (two equations, two unknowns)
Replacing ui1 0 and vi1 0 in the equations
yields
continue
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Newton-Raphson (two equations, two unknowns)
Solving the equations algebraically yields
Alternatively, we may solve for xi1 and yi1
using well-known methods for solving systems of
linear equations
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Newton-Raphson Example

• To solve

First evaluate
With x0 1.5, y0 3.5, we have
continue
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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.
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Newton-Raphson (n equations, n unknowns)
Want to find xi (i 1, 2, , n) that satisfy
From 1st-Order Taylor Series Approximation, we
have
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Newton-Raphson (n equations, n unknowns)
For each k 0, 1, 2, , n, setting fk,i1 0
yields
These equations can be expressed in matrix form as
where
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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.