SLIDE 1 Numerical integration
Numerical and Scientific Computing with Applications David F . Gleich CS 314, Purdue November 8, 2016
HW Due, Odds & Ends of important topics
Next class
QUIZ! More numerical integration!
Next next class In this class you should learn:
problem is fundamentally ill-conditioned
- Derivatives of polynomial
approximations
polynomials
SLIDE 2
Ill conditioning & numerical differentiation
Juliabox Demo!
SLIDE 3
The take-away intuition
Numerical differentiation is sensitive to errors in the function values. But the sensitivity seems mostly proportional to the magnitude of the perturbation. It doesn’t grow “exponentially” Not especially ill-conditioned away from singularities
SLIDE 4 Multivariate functions
−5 5 −5 5 0.2 0.4 0.6 0.8 1 x y
f(x, y) = 1 1 + x2 + y2
SLIDE 5
Multivariate polynomials
A bi-variate (2 variable) quadratic has 9 unknown parameters A bi-variate (2 variable) cubic has 16 unknown parameters A tri-variate (3 variable) quadratic has 27 unknown parameters A tri-variate (3 variable) cubic has ? unknown parameters
f(x, y) = c2,2x2y2 + c1,2xy2 + c0,2y2 + c2,1x2y + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
SLIDE 6
Degree of multivariate polynomials
x2 y2 has degree “four” x y2 has degree “three” the degree of a multivar poly is the degree of the largest term
SLIDE 7
Degree of multivariate polynomials
x2 y2 has degree “four” x y2 has degree “three” the degree of a multivar poly is the degree of the largest term
SLIDE 8
Degree of multivariate polynomials
x2 y2 has degree “four” x y2 has degree “three” the degree of a multivar poly is the degree of the largest term
f(x, y) = c2,2x2y2 + c1,2xy2 + c0,2y2 + c2,1x2y + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
SLIDE 9
Degree of multivariate polynomials
x2 y2 has degree “four” x y2 has degree “three” the degree of a multivar poly is the degree of the largest term
f(x, y) = c2,2x2y2 + c1,2xy2 + c0,2y2 + c2,1x2y + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
f(x, y) = c0,2y2 + + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
SLIDE 10
Quiz
Write down the equations for a multi-linear function in three dimensions: (1) where all degrees are less than or equal to 1 (2) where all “linear” terms of present
f(x, y) = c2,2x2y2 + c1,2xy2 + c0,2y2 + c2,1x2y + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
f(x, y) = c0,2y2 + + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
SLIDE 11
SLIDE 12 Fitting multivariate polynomials
… not nice to write down in general …
- Saniee. “A simple form of the multivariate Lagrange interpolant”
SIAM J. Undergraduate Research Online, 2007.
f(x, y) = c0,2y2 + + c1,1xy + c0,1y + c2,0x2 + c1,0x + c0,0
SLIDE 13 An easier special case
If we have data from
repeated grid then we can fit a sum of 1d polynomials “tensor product constructions”
x1 x2 x3 x4 x5 x6 y1 y2 y3 y4 y5 y6 y7
p(x, y) = X zijϕx
i (x)ϕy j (y)
SLIDE 14
The big problem
If we have an m dimensional function And we want an n degree interpolant We need (n+1)m samples of our function. “quadratic” in 10 dimensions – 310 samples “quadratic” in 100 dimensions – 3100 samples Exponential growth or “curse of dimensionality”
