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Numerical Integration and DifferentiationComputational Economics

Dietmar Maringer

WWZ, University of Basel

Spring 2012

D Maringer,Computational Economics Numerical Integration and Differentiation (1)


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Finite Differences for Numerical Differentiation

basic concepts

difference quotient: yx=

f(x0+x)f(x0)x

derivative: dydxf(x) limx0 yx

numerical approximation offwith centraldifference

f(x)

f(x+)f(x)(x+) (x)

= f(x+)f(x)2

where 0

xx

x+

x

f(x)x+

f(x+)

multi-dimensional functions

gradient: list of all first order derivatives

f= fx1

fxn

f(x1+,x2,...,xn)f(x1,x2,...,xn)2

f(x1,x2,...,xn+)f(x1,x2,...,xn)2



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second derivative for one-dimensional functions

since second derivative = derivative of first derivative, compute by

iteratively applying definitions:

f(x) f(x+)f(x )(x+) (x)

=

f(x++)f(x+)2

f(x+)f(x)2

(x+) (x)= f(x+2)+f(x2)2f(x)

(2)2

= f(x+)+f(x)2f(x)2

where = 2likewise for higher derivatives



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second derivative for multi-dimensional functions

Hessian matrix multi-dimensional functions: matrix of all second

order derivatives

H=

2f

x21

2fx1xn

...

2f

xixj

...

2f

xnx1

2f

x2n

usually, Hessian is symmetric, i.e.,

2fxixj

= 2f

xjxi



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numerical approximation of the Hessian

vary values in dimensionsiandj(leaving all other values unchanged)

and evaluate points

xj xj xj+xi f f f+xi f

f f+xi+ f+ f+ f++

diagonal elements of the Hessian matrix

2f

x2i

f++f2f

2 alternative 1

12

f++f2f

2 + f

+++f+2f+2

alternative 2

off-diagonal elements of the Hessian matrix (cross-derivatives)

2f

xixj

12

f++fff+

2 + f

++ff+f2

alternative 1

12

f+f++f+f+

2 + f

+fff2

alternative 2

14

f+++ff+f+2 average alt.s 1 and 2



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Application Example

local approximation with Taylor Series Approximation

approximation for region around a given valuex0

approximate function value atx= x0+with a polynomial

f(x)=f(x0+)=f(x0)+f(x0)+

2

2 f(x0)+

3

6 f(x0)+

+ +

n

n!f(n)

(x0)+Rn

remainderRn < |xx0|n+1


i l i i h id l


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Numerical Integration with Trapezoid Rule

basic idea

assume function is approximately piecewise linear

area underneath each line segment is a trapezoidintegral is the sum of all trapezoids areas

=x1

=x4x2 x3

f2

f3f2+f3

2

wa b

w f2+f32


N i l I i i h T id R l


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Numerical Integration with Trapezoid Rule

how solve a problem likeb

af(x)dx

split range intonsegments of equal widthw= (ba)/ncompute borders of segments,xi = a+ (i1)wfori= 1..(n+1)approximation for area of one segment:

xi+1xi

f(x)dxwf(xi)+f(xi+1)2for the entire range (withfi =f(xi))

b

af(x)dx= n

i=1

xi+1

xi

f(x)dx

n

i=1w fi+fi+1

2 =w

f12+f2+ +fn+

fn+12

more generally (when segments have different width):ba

f(x)dxn

i=1(xi+1xi)

fi+fi+12

extension: Simpsons rule (piecewise quadratic functions) D Maringer,Computational Economics Numerical Integration and Differentiation (8)

N i l I t ti ith M t C l Si l ti


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Numerical Integration with Monte Carlo Simulation

some (very) simple examples

f(x)= 12

x=

2

0f(x)dx=? f(x)= sin

|x| 5

|x|=

3

1f(x)dx=?

Estimating the Value ofvia MCS

A = (2r)2

A = r2AA

= r2

4r2=

4

== 4AA

y

x

r

1 1

1

1


dditi l lit t


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additional literature

Gilli, M., Maringer, D., and Schumann, E. (2011).

Numerical Methods and Optimization in Finance.

Academic Press.Judd, K. L. (1998).

Numerical Methods in Economics.

MIT Press.

Miranda, M. J. and Fackler, P. L. (2002).

Applied Computational Economics and Finance.MIT Press.


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