CSE 541 Numerical Methods

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CSE 541Numerical Methods

• Mathematical Preliminaries and Algorithms

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Review of Functions

• A function is a mapping from a set of values
  (called the domain) to another set of values
  (called the range)
• A 1D function, y f(x) or expressed as (x, f(x))
• x is the independent variable and y is dependent
  variable
• Higher dimensional functions, z f(x, y)
• We will stick with 1D functions in lecture for
  the most part

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Algorithm Function Extrema

• Take a function f(x) and any point t on the
  domain, give me an algorithm that will tell
  determine if f(t) is an extrema

y
f(x)
x
t
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Review of Functions

• First derivative, f (x)
• How fast the function changes
• Slope
• Mathematical definition
• Extrema of a function occur where f (x) 0

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AlgorithmFunction Minimum or Maximum

• Take a function f(x) and any point t on the
  domain, give me an algorithm that will tell
  determine if f(t) is a function min or max

y
f(x)
x
t
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Review of Functions

• Second derivative, f (x)
• Acceleration
• Indicates curvature
• Determines whether the function at x is a minimum
  or maximum

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Tangents (Gradients) and Normals

• Assume a 2nd degree function
• Geometric formulation of the first derivative
  Tangent line
• A linear approximation to the curve at t is l(x)
  b f(t)x
• l(x) intersects the curve at t, so it
  interpolates the function

f (t)
N -1/f (t)
y
f(x)
x
t
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Higher Dimensional Functions

• A surface is a 2D function in 3D
• Surface (x, y, f(x,y) )
• A volume (hyper-surface) is a 3D function in 4D
• Volume (x, y, z, f(x,y,z) )

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Partial Derivatives

• Keep the other dimensions constant

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Tangent (Gradient)

• Assume a 2nd degree function
• Tangent at point (s, t) is a linear slope, i.e.
  a plane
• axbyczd 0

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Tangent Plane Construction

• Assume a 2nd degree function
• Give me a geometric algorithm to construct the
  tangent plane at point (s, t)

Images courtesy of TJ Murphy http//www.math.ou.e
du/tjmurphy/Teaching/2443/TangentPlane/TangentPla
ne.html
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Tangent Plane Construction
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Tangent Plane Construction
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Normal

• The normal to the plane is
• A vector perpendicular to the tangent plane

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The Class of Polynomials

• Specific functions of the form
• For many polynomials, the latter coefficients are
  zero. For example
• p(x) 3x25x3

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Algorithm Efficiency

• Suppose we want to evaluate this polynomial at x
  2
• f(x) 2x4 x3 3x2 x1 2
• How many mults are required to compute f(2)?
• Re-write the expression as
• f(x) (((2x 1)x 3)x 1)x 2
• This is called Horners rule or method.

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Errors in Calculcations

• A program implementation of a calculation will
  introduce errors, e.g.
• We may know the true solution, e.g. f(x) x2

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Errors in Calculations

• Error
• How far away am I from the true solution?
• Simple definition a b , where a is the
  true value and b is an approximation
• We may only be able to compute an approximation
• Approxiate Pi?
• Determining an error is not easy in many cases
• May be able to find an error bound, a b lt e

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Absolute vs Relative Errors

• Are all errors the same?
• a 1,000,000 and b 1,000,000.01, error .01
• vs
• a .02 and b .01, error .01
• Absolute error a b
• Relative error a b / a
• Scale factor brings things into perspective