cos323 f11 lecture14 integration - Princeton University Computer … · 2011. 12. 4. · • Polynomial interpolation ... integral with a single polynomial and evenly-spaced points

Integration

COS 323

Last time

•  FFT Wrap-up •  Polynomial interpolation

– Monomial basis, Lagrange interpolation

•  Intro to numerical integration

Today

•  Review numerical integration – Formulation, central considerations

•  Interpolatory Quadrature – Review formulation; error analysis

•  Newton-Cotes Quadrature – Midpoint, Trapezoid, Simpson’s Rule

•  Error analysis for trapezoid, midpoint rules •  Richardson Extrapolation •  Gaussian Quadrature

Numerical Integration Problems

•  Basic 1D numerical integration – Given ability to evaluate f (x) for any x, find

– Goal: best accuracy with fewest samples – Classic problem – even analytic functions not

necessarily integrable in closed form

Quadrature

1.  Sample f(x) at a set of points 2.  Approximate by a friendly function 3.  Integrate approximating function

•  Choices: – Which approximating function? – Which sampling points? (“nodes”)

•  Even vs. uneven spacing?

– Fit single function vs. multiple (piecewise)?

Today’s topics

•  Newton-Cotes methods: Estimating function over integral with a single polynomial and evenly-spaced points (subset of interpolatory quadrature)

•  Piecewise application of several polynomials •  Richardson extrapolation: iteratively improving a

result

•  Adaptive quadrature: choosing node positions according to function

•  Gaussian quadrature: “best” polynonial interpolation using optimally-chosen points (independent of function)

Trapezoidal Rule

•  Approximate function by trapezoid

•  Approximate integral by parabola through three points

Simpson’s Rule

Quadrature: General Formulation

•  n-point qudarature rule:

•  Open if a < x1 and xn < b, closed if a = x1, xn = b

Quadrature: General Formulation

•  n-point qudarature rule:

•  Open if a < x1 and xn < b, closed if a = x1, xn = b

Interpolatory Quadrature

•  Use n points to approximate f using a polynomial of degree n-1 – Yields a quadrature rule “of degree n-1”

•  For a “smooth” function f:

Newton-Cotes family of methods

•  n-point Newton-Cotes: n evenly-spaced points in interval (open or closed)

•  1 point, open interval: midpoint rule

•  2 points, closed interval: trapezoidal rule

•  3 points, closed interval: Simpson’s rule

•  Approximate integral by parabola through three points

Simpson’s Rule

Midpoint rule error analysis

Midpoint rule error analysis

•  In general, error for a single segment proportional to h3 – Note that only odd powers of h (even terms of

Taylor expansion) remain in error expression

Midpoint vs Trapezoid

1.  Error(T) > Error(M) (factor of 2!) 2.  Halving interval error decreases by 1/8 3.  Observation 2: Can use M(f) and T(f) to

learn about error

Simpson’s Rule Error

•  Better accuracy than midpoint or trapezoid – Global error O(h4), exact for cubic (!) functions

•  Higher-order polynomials (Newton-Cotes): – Global error O(hk+1) for k odd, O(hk+2) for k even – Fits polynomial of degree k for k points, k odd – Or polynomial of degree k-1 for k points, k even

Surprise benefts of odd-point rules

•  Errors cancel exactly if true function is polynomial of degree n. (only if n odd!)

•  Simpson’s rule is usually preferred over trapezoid & midpoint

Why not just keep increasing n?

€

Qn ( ˆ f ) −Qn ( f ) ≤ wii=1n

∑( ) ˆ f − f ∞where wi = b − a∑i.e., arbitrarily bad conditioning can result if some wi negative

Newton-Cotes:

€

n ≥11 means some wi < 0

and wi →∞ as n∑ →∞

Composite Quadrature

Extended/Composite/Multiple-Application Trapezoidal Rule

Divide into segments of width h, piecewise trapezoidal approximation

Extended Midpoint Rule

Divide into segments of width h:

Error analysis

•  Halving interval length reduces error by 1/8 •  BUT twice as many intervals needed effect

is error reduction by ¼

•  Composite midpoint or trapezoid: O(h2) – Compare to h3 for a single trapezoid – Midpoint still better constant terms than trapezoid

•  Composite Simpson’s: O(h4)

How small should h be?

•  Can’t necessarily compute f(2) , so can’t compute error directly

•  Can estimate error: 1.  I(h1)= quadrature with width h1 2.  I(h2)= quadrature with width h2=.5h1 3.  Estimate error ≅ I(h2) – I(h1)

Progressive Qudrature

•  Re-use nodes from Qn1 to compute Qn2 •  Trapezoid: Halve intervals •  Midpoint:

– Can cut interval into thirds:

Caution: Round-off error lurks

Adaptive Quadrature

Richardson Extrapolation &

Romberg Interpolation

Using knowledge of error to improve a solution

€

I = I h( )+ E h( )I(h1) + E(h1) = I(h2) + E(h2)

E ≅ − b − a12

h2 ʹ′ ʹ′ f

E(h1) ≅ E(h2)h1h2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

If using trapezoidal rule and h2 =12

h1 :

I ≅ 43

I(h2) −13

I(h1)

and the new estimate has error O(h4 ) insetead of O(h2)

Example

€

Integrate f (x) = 0.2 + 25x − 200x 2 + 675x 3 − 900x 4 + 400x 5

from a = 0 to b = 0.8 (true value ≈1.640533)# of trapezoids

h Integral

1 .8 0.1728 2 .4 1.0688 4 .2 1.4848

€

ˆ I 1 =43

(1.0688) − 13

(0.1728) =1.367467

ˆ I 2 =43

(1.4848) − 13

(1.0688) =1.623467

Romberg Interpolation

•  Can repeat:

€

Fh

Fh / 2

Fh / 4

Fh / 8O(h2)k =1

O(h4 )k = 2

O(h6)k = 3

O(h8)k = 4

€

I j,k ≈4 k−1I j+1,k−1 − I j ,k−1

4 k−1 −1

Richardson Extrapolation

•  This treats the approximation as a function of h and “extrapolates” the result to h=0

•  (When using trapezoid and halving intervals, this is Romberg Interpolation)

Gaussian Quadrature

Gaussian Quadrature

•  When free to choose node locations, yields degree 2n-1 for n nodes!

•  Apply n-point G.Q. nodes from “rules”

Example: 2-point Gaussian Quadrature

•  Integration equation:

•  When x1, x2 are fixed, use moment equations to derive polynomial of degree 1:

Example: 2-point Gaussian Quadrature

Solution:

•  Resulting polynomial is of degree 3! •  Tables exist with weights and nodes for GQ

of degree n, integrated over [-1,1]

•  In practice, transform function to rescale x in [a, b] to [-1, 1]

Other Techniques

Limits at Infinity

•  Usual trick: change of variables

•  Works with a, b same sign, one of them infinite – Otherwise, split into multiple pieces

•  Also requires f to decrease faster than 1/x2 – Else need different change of variables, if

possible!

€

f (x)dxa

b

∫ = 1t 2 f 1t( )dt1/ b

1/ a

∫

Example: Standard normal distribution

€

=12π−∞

z∫ e−x 2 / 2dx

Discontinuities

•  All the above error analyses assumed nice (continuous, differentiable) functions

•  In the presence of a discontinuity, all methods revert to accuracy proportional to h –  In general, if the k-th order derivative is discontinuous,

can do no better than O(hk+1)

Discussion

•  Newton-Cotes: –  Simple, often effective –  Not highest degree possible for given number of nodes –  Progressive, can be used for adaptive

•  Gauss: –  Not for tabular data, requires linear change of function, best

degree for given number of nodes, can get very high accuracy

•  Romberg: –  Richardson based on trapezoid; not for tabular data, can get

very high accuracy

Matlab functions

•  trapz: trapezoidal with consistent spacing •  polyint: integrate polynomial analytically •  quad: Adaptive Simpson quadrature •  quadgk: Adaptive Gauss-Kronrod •  Also: quadl, cumtrapz, dblquad, triplequad