Home >Documents >Iterative methods for nonlinear systems of equations: an introduction · PDF file...

# Iterative methods for nonlinear systems of equations: an introduction · PDF file...

Date post:21-Apr-2020
Category:
View:0 times
Transcript:
• Iterative methods for nonlinear systems of equations:

an introduction

Laboratori de Càlcul Numèric (LaCàN) Dep. de Matemàtica Aplicada III

Universitat Politècnica de Catalunya www-lacan.upc.es

• 2

Contents

§ Problem statement § Motivation § Functional iteration § Method of direct iteration § Picard’s method § Newton’s method § Quasi-Newton methods

• 3

Problem statement

§ Starting point

§ General form of nonlinear system

(1 nonlinear equation with 1 unknown)

(linear system of order n)

f transforms vectors into vectors

α is a zero of f if

• 4

Problem statement

In more detail:

In general, all components of f are nonlinear w.r.t. all components of x

• 5

Problem statement

§ Particular form of nonlinear system

with and

Similar structure to linear system

Can be (trivially) transformed into the general form:

• 6

Motivation

Mathematical models in engineering Linear models § Response of physical system proportional to external actions § Simple models § A first approximation to the real behaviour

Examples: linear systems of equations; linear PDEs

¬ Nonlinear models § No proportionality between actions and response § More complex models § More realistic description of real beaviour Examples: nonlinear systems of equations; nonlinear PDEs

• 7

Functional iteration

§ Analogy with root finding in 1-D: 1-D problem n-D problem

§ Consistency: function φ must verify

(zeros of f) (fixed points of φ)

Nonlinear equation(s)

Initial approximation

Iterative scheme

• 8

Functional iteration

§ Convergence: contractive mapping theorem

Let φ: D D, D a closed subset of R . If there exists λ ∈ [0,1) such that

then: (a) there exists a unique fixed point α of φ in D.

(b) for any initial approximation x in D, the sequence {x } generated by x = φ(x ) remains in D and converges linearly to α with constant λ.

n

0 k kk+1

• 9

Method of direct iteration (or successive approximations)

Advantages § Very simple technique (evaluate f once per iteration) Drawbacks § Contractivity of φ not guaranteed § Convergence is typically linear (if it converges!)

Problem in general form

Iteration function

Iteration scheme

• 10

Method of direct iteration (or successive approximations)

• 11

Picard’s method (or secant matrix method)

Problem in particular form

If matrix is inversible,

define the iteration function

so the iteration scheme is

Attention: do not invert matrix!

• 12

Picard’s method (or secant matrix method)

§ Solve one linear system per iteration

§ Matrix A(x) and vector b(x) one iteration behind

If matrix is inversible,

Practical algorithm

• 13

Picard’s method (or secant matrix method)

Matrix A(x) is a secant matrix

• 14

Picard’s method (or secant matrix method)

Advantages § If A(x) has a special structure (e.g. banded SPD),

it can be exploited when solving the linear systems

Drawbacks § Matrix A(x) may be singular for some x

§ Convergence is typically linear (if it converges!)

§ Computational cost: matrix A(x) and vector b(x) change at every iteration

• 15

Newton’s method (or Newton-Raphson’s method)

Problem in general form

Correction of non-converged approximation:

First-order Taylor’s series expansion:

• 16

Newton’s method (or Newton-Raphson’s method)

Jacobian matrix

Involves the computation of derivatives

• 17

Newton’s method (or Newton-Raphson’s method)

Practical algorithm

Iteration function is

Jacobian matrix not inverted in practice!

• 18

Newton’s method (or Newton-Raphson’s method)

• 19

Newton’s method (or Newton-Raphson’s method)

For problem in particular form, the iterative scheme is

similar to iterative schemes for linear systems

The Jacobian matrix does not retain structure of A(x)

• 20

Newton’s method (or Newton-Raphson’s method)

Drawbacks § Matrix J(x) may be singular for some x

§ Computational cost: at every iteration, (1) compute matrix J(x) and vector f(x) and (2) solve linear system

§ If A(x) has a special structure (e.g. banded SPD), it is lost when computing J(x)

• 21

Quasi-Newton methods § Secant method in 1-D

Similar to Newton’s method

with tangent approximated

by secant defined by the last two iterations k-1 and k

• 22

Quasi-Newton methods

• 23

Quasi-Newton methods § Extension to nonlinear systems

Jacobian matrix is approximated by a secant matrix

defined by the last two iterations k-1 and k

n unknowns and only n equations additional conditions on matrix S required

2

Embed Size (px)
Recommended

Documents

Documents

Documents

Documents

Documents

Documents