Numerical Analysis

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Numerical Analysis

• The information is not in the course textbook.
• The last couple of decades have seen research
  focused on non-numeric research
• Historically, numerical analysis was one of the
  first areas of interest in computer science
  including
• Interpolation and extrapolation algorithms
• Solving algebraic and transcendental equations
• Simultaneous linear equations
• Initial-value, characteristic-value, and
  boundary-value problems
• Partial differential equations

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Numerical analysis

• Approximate computations are greatly affected by
  errors
• Numerical errors arise from different causes and
  affect results in different ways
• Round-off errors are due to the representation of
  a number by a finite number of decimal digits.
  (Clearly, an irrational number cannot be
  precisely represented by a finite of digits)
• Truncation errors are when a function f(x) is
  represented by an infinite series but the series
  is cut off after a finite number of terms.
• Accumulated errors are errors that build up
  during some sequence of operations. If the rate
  of accumulation decreases, the error is bounded
  and said to be stable. If it is unstable, it
  renders the solution meaningless.
• Absolute and relative errors need to be taken
  into account.

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Numerical Analysis

• Since we are dealing with approximations to the
  real values, it is important to distinguish
  between absolute error and relative error, of
  representing a number ? by its approximation
  ?absolute error ? - ?,
  ? - ?relative error ----------
  ?

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Numerical Analysis-finding roots

• Methods of finding roots of an equation given as
  f(x) 0. We wish to determine x such that the
  above equation is true.
• Quadratic equation only for 2 roots
• Bisection method converges slowly
• Newton-Raphson converges rapidly but may
  diverge.

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Quadratic Equation

• This is well known from algebra
• ax2 bx c 0
• has roots at

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Quadratic Equation

• In general, we cannot solve equations exactly and
  need approximation algorithms to do so.
• We cannot solve square roots exactly
• We do not have formulas for polynomials above
  degree 4, but even the 3rd and 4th degree
  equations are too cumbersome to be of value
• In fact, Paolo Ruffini (1765-1822) in 1799
  discovered that there can be no general formula
  for their roots involving only the coefficients,
  arithmetic operations, and taking square roots.

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Bisection Method

• This method is based upon the observation that a
  graph of a continuous function must intersect
  with the x-axis between two points a and b at
  least once if the functions values have opposite
  signs at these two points.
• See the next slide for an illustration

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Bisection Method

• Since we cannot expect the algorithm to stumble
  on the exact root, we need a criterion for
  stopping the algorithm.
• We can stop the algorithm after the interval
  a,b bracketing some root x becomes so small
  that we can guarantee that the absolute error of
  approximating the root x by xn, the middle point
  of the interval, is smaller than some small
  pre-selected number, say ?gt0. Since xn is the
  midpoint of a,b and x lies within this
  interval, we have

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• Hence, we can stop the algorithm as soon as

or, equivalently, when
11
Bisection Method

• The previous slide implies that the sequence of
  approximations can be made as close to root x as
  we with by choosing n large enough. In other
  words, xn converges to x.
• Note, however, that because any digital computer
  represents extremely small values by zero, the
  convergence assertion is true in theory, but not
  necessarily in practice.
• In fact, if we choose ? below a certain
  machine-dependent threshold, the algorithm may
  never stop!
• Another potential complication is round-off
  errors in evaluating the function in question.
  Therefore, in practice, we usually limit the
  bisection algorithm to a certain maximum number
  of iterations.

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Newton-Raphson

• This is one of the most important general
  algorithms for solving equations. The general
  analytical formula looks like

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Newton-Raphson

• In most cases, Newtons algorithm guarantees
  convergence of the sequence if an initial
  approximation is close enough to the root.
  However, as shown on the previous slide, it may
  not converge for certain situations.
• Newtons method generally converges much faster
  than bisection.
• Note that Newtons method requires an evaluation
  of the function and the derivative of the
  function whereas bisection requires only an
  evaluation of the function.
• A possible criterion for stopping Newtons
  algorithm is simply when two successive guesses
  are lt ? apart, or when xn xn-1 lt ?