Introduction to Numerical Methods

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Introduction to Numerical Methods

• Andreas Gürtler
• Hans Maassen (maassen_at_math.ru.nl)tel. 52991

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Plan for today

• Introduction
• Errors
• Representation of real numbers in a computer
• When can we solve a problem?
• A first numerical methodBisection method for
  root-finding (solving f(x)0)
• Practical part Implementation of the bisection
  method

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What are numerical methods for?

• Many problems can be formulated in simple
  mathematical equationsThis does not mean, that
  they are solved easily!
• For any application, you need numbers?
  Numerical Methods!
• Needed for most basic things exp(3),sqrt(7),sin(
  42), log(5) , ?
• Often, modeling and numerical calculations can
  help in design, construction, safety
• Note many everydays problems are so complicated
  that they cannot be solved yet ? Efficiency is
  crucial

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Numerics is about numbers

• Numerical methods Numerical approximation of
  solutions to understood problems
• Numerical representation of real numbers has
  far-reaching consequences
• Two main objectives
• quantify errorsApproximation without error
  estimation is useless
• increase efficiencySolutions which take years or
  need more resources that you have are useless
• Nowadays, many fields depend on numerics

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• Errors

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Numerical errors

• Roundoff error finite precisionnumerical
  calculations are almost always approximations
• Truncation error a calculation has to
  stopExamples
• Approximation (e.g. finite Taylor series)
• Discretization
• It is crucial to know when to stop (i.e. when a
  calculations is converged!). To check this,
  change parameters (e.g. step size, number of
  basis states) and check result.
• modeling error

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Truncation errors

• Truncation errors are problem specific
• Often, every step involves an approximation,e.g.
  a finite Taylor series
• The truncation errors accumulate
• Often, truncation errors can be calculated

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Roundoff errors

• Precision of representation of numbers is
  finite - errors accumulate
• a real number x can be represented as fl(x)
  x(1?) floating point computer
  representation fl(x)-x ?x
  absolute error (often also ?x) fl(x)-x/x
  ? relative error

• Example Logistic Mapxi1r xi (1-xi)
• x0 0.7 r4
• single and double precision

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• Computer representation of numbers

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back to Matlab data types

• Real numbers floating point representation
  double (64 bit) standard! single (32
  bit) less precision, half memory
• Integer signed and unsigned (8,16,32
  bit)auint8(255), bint16(32767),
  cint32(231-1)Integers use less space,
  calculations are precise!
• Complexw23i vcomplex(x,y) x,y can be
  matrices!
• Boolean true (1) or false (0) Strings
  s'Hello World'
• special valuesinf, -inf, NaN Infinity,
  Not-a-Numbercheck with isinf(x), isnan(x)

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floating point caveats

• not all real numbers can be represented as
  FP0.10.10.10.10.10.10.10.10.10.1-1
  -1E-16
• Floating point calculations are not
  precise!sin(pi) is not 0, 1E201-1E20 is not
  1
• never compare two floats (ab) directlytry
  100(1/3)/100 1/3 FALSE!!! use
  abs(a-b)lteps (eps2.2E-16)
• be careful with mixing integer and
  FPiint32(1)i/2 produces 1 as i/2 stays
  integer!i/3 produces 0 ! This is much more
  dangerous in C and FORTRAN etc. Solution
  explicit type conversion double(i)/2 produce
  s 0.5

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• What can we solvenumerically?

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What can we solve

• Suppose we want to evaluate f(x) with perfect
  algorithm
• we have FP number x?x with error ?x?f(x)
  f(x?x)-f(x) ? f(x)?x (if f differentiable)
• relative error
• Definition condition number
• ? gtgt1 problem ill-conditioned
• ??small problem well-conditioned

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well- and ill-conditioned methods

• Lets try a simple calculation 99-70sqrt(2)
  (?0.00505)
• suppose we have 1.4 as approximation for ?2
• We have 2 mathematically equivalent methods
• f1 99-70?2 f1(1.4) 1
• f2 1/(9970?2) f2(1.4) ? 0.0051
• Condition numbers f1(x) 99-70 x ????2? ?
  20000f1(x) 1/(9970 x) ????2? ? 0.5

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what happened?

• f1 99-70?2
• f2 1/(9970?2)
• Condition number of subtraction,
  additionf(x)x-a ?? -x/(x-a) ill-condition
  ed for x-a0f(x)xa ?? x/(xa)
  ill-conditioned for xa0
• Condition number for multiplication,
  divisionf(x)ax ? xa/(ax) 1
  f(x)1/x ? xx-2/(x-1) 1 well-conditioned
  

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numerical disasters

• Patriot system hit by SCUD missile
• position predicted from time and velocity
• the system up-time in 1/10 of a secondwas
  converted to seconds using 24bit precision (by
  multiplying with 1/10)
• 1/10 has non-terminating binary expansion
• after 100h, the error accumulated to 0.34s
• the SCUD travels 1600 m/s so it travels gt500m in
  this time
• Ariane 5
• a 64bit FP number containing the horizontal
  velocity was converted to 16bit signed integer
• range overflow followed
• from
• http//ta.twi.tudelft.nl/nw/users/vuik/wi211/disas
  ters.html

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Root finding a standard numerical problem

• A common task is to solve an equation f(x)0
• task find the zero-crossing (root) of f(x)
• Pitfalls f(x) might have
• no roots
• extrema
• many roots
• singularities
• roots which are not FP numbers
• roots close to 0 or at very large x
• There are different methods with advantages and
  disadvantages

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convergence

• Bisection always converges, but maybe slow
• f ? Ca,b and f(a)f(b)lt0 then pn
  converges to a root p of f andpn-p ?
  (b-a)/2n
• Proof?n ?1 bn-an (b-a)/2n-1 and p ?
  (an,bn)
• convergence is linear

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order of convergence
Let converge to p with
if positive ???? exist with
then pn converges with order ??1 linear
convergence?2 quadratic convergence
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Exercise

• Implement the Bisection method in Matlab
• use it to solvex3 ln(1x)
  (xgt0)

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