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
Download:0 times
Share this document with a friend
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)

    Advantages § Convergence is quadratic (for J(α) not singular)

    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

Click here to load reader

Reader Image
Embed Size (px)
Recommended