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M. Dumbser
1 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Lecture on Numerical Analysis
Dr.-Ing. Michael Dumbser
24 / 09 / 2008
M. Dumbser
2 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Numerical Integration (Quadrature) of Functions - Motivation
a b x
f
b
a
n
ii
n
i
x
x
dhxfhdxxfdxxfi
i1
1
01
)()()(1
b
a
dxxf ?)(
h
n
abh
hiaxi
Task: compute approximately
Solution strategy:
• Divide interval [a;b] into n smaller subintervals
• Approximate f(x) by interpolation polynomials on the subintervals, e.g. using Lagrange interpolation
• Integrate these polynomials exactly on each subinterval and sum up
• So-called Newton-Cotes formulae
ni ,...1,0
M. Dumbser
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Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Transformation of the Integration Interval
1
0
1
0
)(~
:)()(1
dfhdhxfhdxxf i
x
x
i
i
The computation of numerical quadrature formulae for each sub-interval can be technically considerably simplified using the following variable substitution:
hxx i ii xxh 1 hd
dx
Therefore, it is sufficient, without the loss of generality, to consider from now on the case of numerical integration in the reference interval [0;1].
...)(~1
0
df
M. Dumbser
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Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Newton-Cotes Formulae
The integral of f(x) is approximated in the following steps:
1. First, the function f(x) is interpolated by a polynomial of degree k insideeach sub-interval. The interpolation points are distributed in an equidistantmanner in each sub-interval.
2. Second, the interpolation polynomial is integrated analytically.
3. Steps 1 and 2 produce an approximation of the integral of f(x) in termsof the function values fi at the interpolation points and the step size h.
M. Dumbser
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Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Trapezium or Trapezoid Rule
a b
f
hn
abh
hiaxi 1
Use linear interpolation polynomials, i.e. polynomials of degree one in the subintervals [xi;xi+1] [0;1]:
01
0
10
111
ii ffPf
1
1
0
1 2
1
2
1 ii ffdP
11 2
1
ii
x
x
ffh
dxxPi
i
x
11 1 ii ffP
M. Dumbser
6 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Simpson Rule (Originally Discovered by Johannes Kepler in 1615)
a b
f
hn
abh
hiaxi 1
Use quadratic interpolation polynomials, i.e. polynomials of degree two:
2
1
21
121
21
21
21
2
101
0
10
10
100
1
21
ii
i
ff
fPf
1
1
0
2 6
1
3
2
6
121 iii fffdP 12
21
1
46
iii
x
x
fffh
dxxPi
i
x
2
12
22
24
132
21 ii
i
ff
fP
M. Dumbser
7 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The 3/8 Rule
Use cubic interpolation polynomials, i.e. polynomials of degree three:
3
231
32
31
132
31
32
32
31
31
32
31
31
32
32
31
32
31
3
1101
0
10
10
10
10
1000
1
32
31
ii
ii
ff
ffPf
1
1
0
3 8
1
8
3
8
3
8
132
31 iiii ffffdP
3223
129
312
343
227
322
353
227
92
91123
29
3
32
312
ii
ii
ff
ffP
M. Dumbser
8 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Isaac Newton
Sir Isaac Newton - (* 4. January 1643 in Woolsthorpe; † 31. March 1727 in Kensington)
• Physicist, Mathematician, Astronomer and Philosopher
• Together with Leibniz, Newton is one of the inventors of infinitesimal calculus (differentiation and integration)
• 1667: Fellow of Trinity College, Cambridge
• 1687: Philosophiae Naturalis Principia Mathematica. Newton discovered the universal law of gravitation and the laws of motion of classical mechanics
• 1704: Opticks. A corpuscular theory of light.
• From 1703 president of the Royal Society
• Buried in Westminster abbey in London
M. Dumbser
9 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
The previously derived formulae were all of the form
and were very easy to obtain since an equidistant spacing of the nodes was imposed and only the weights j had to be computed. The aim of the Gaussian quadrature formulae is now to obtain an optimal quadrature formula with a given number of points by making also the nodes an unknown in the derivation procedure of the quadrature formula and to come up with an optimal set of nodes xj and weights wj.
An explicit construction strategy for Gaussian integration formulae:
(1) Using M quadrature points, we have M unknowns for the positions and also M unknowns for the weights, i.e. a total of 2M unknowns.
(2) We need 2M equations to determine uniquely the 2M unknowns.
(3) The equations are obtained by requiring that the integration formula is exact for polynomials from degree 0 up to degree 2M-1 !
jM
jj fdf
1
1
01
1
M
jj
M. Dumbser
10 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
This means we have 2M equations of the form to solve for j and j.
For Pi(), any polynomial of degree i can be used, in particular also the monomial i.
ji
M
jji PdP
1
1
0
12...1,0 Mj
Example 1: One integration point, i.e. M = 1, leading to the two equations:
111 101
1
0
1
0
0
j
M
jjPddP
1111
1
0
1
0
1 2
1
j
M
jjPddP
11
112
1
2
1,1 11
M. Dumbser
11 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
Example 2: Two integration points, i.e. M = 2, leading to the 4 equations:
21
1
0
11 d
2211
1
0 2
1 d
36
1
2
1,3
6
1
2
1,
2
12121
222
211
1
0
2
3
1 d
322
311
1
0
3
4
1 d
M. Dumbser
12 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
A more efficient and more general way of obtaining the Gaussian quadrature formulaemakes use of so-called orthogonal polynomials Li(), which are the so-called Legendrepolynomials.
1
0
)()(, dgfgf
The set of polynomials Li() is called orthogonal, if it satisfies the relation
ji
jiLL
iji if
if0,
First, we define the scalar-product of two functions f and g as follows:
With this scalar product available, we can define the L2 norm of a function f as
fff ,
M. Dumbser
13 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
The polynomials Li() can be constructed via Gram-Schmidt orthogonalization from the monomials M0 = 1, M1 = 2, M3 = 3, … Mn = n as follows:
We first use the analogy of the scalar product of two functions with thescalar product already known for vectors:
i
ii baba
, aaa
,
The Gram-Schmidt orthogonalization then proceeds as follows:
00 : LM
1M
0
0
0
01, L
L
L
LM
0
0
0
0111 ,
L
L
L
LMML
1M
M. Dumbser
14 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
Instead of performing orthogonalization of vectors, we now perform the orthogonalization of functions as follows:
1:)(0 L
00
01011 ,
,)()(
LL
LMLML
00 LM
1M 1M
00
020
11
12122 ,
,
,
,)()(
LL
LML
LL
LMLML
00
030
11
131
22
23233 ,
,
,
,
,
,)()(
LL
LML
LL
LML
LL
LMLML
0
0
0
0111 ,
L
L
L
LMML
0
0
0
01, L
L
L
LM
M. Dumbser
15 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
Using the orthogonality property of the Legendre polynomials, we find that
else0
0if1,1)(
1
0
iLdL ii
The Gaussian quadrature formulae are written as
)()(1
1
0
j
n
jj fdf
If we now apply formula (2) to the integrals given in (1), we obtain the following linear equation system for the weights j, if we suppose the positions j to be known:
(1)
(2)
else0
0 if1)()(
1
1
0
iLdL ji
n
jji (3)
)(with jiijijijLLeL
(3‘)
M. Dumbser
16 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
Theorem:
If the n positions j are given be the n roots of the polynomial Ln ( Ln( j ) = 0 ), and the weights are given by the solution of system (3), then the Gaussian
quadrature rules are exact for polynomials up to degree 2n-1, i.e.
Proof:
Suppose p() is an arbitrary polynomial of the space of polynomials of degree 2n-1, i.e.
Then we can write the polynomial p() as
12)( nPp
)()()()( rqLp n 1)(),( nPrq
)()(1
0
n
kkkLq )()(
1
0
n
kkkLr
n
jjj pdp
1
1
0
)()(
M. Dumbser
17 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
The Gaussian Quadrature Formulae
Proof (continued):
We have
We also have
0
1
0
1
0
,1,)()()()( rqLdrqLdp nn
)()()()(11
jjjn
n
jjj
n
jj rqLp
1
011
)()(n
kjkk
n
jjj
n
jj Lp
01
1
01
)()(
n
jjkj
n
kkj
n
jj Lp
This finishes the proof. QED
M. Dumbser
18 / 18
Analisi NumericaUniversità degli Studi di TrentoDipartimento d‘Ingegneria Civile ed AmbientaleDr.-Ing. Michael Dumbser
Integration of Improper IntegralsIf the integration interval goes to infinity, it can be very useful to change the integration variables use the following substitution:
222
11)(,)(
1111)(
1
1
1
1
1
1 ttftgdttgdt
ttfdt
ttfdxxf
a
b
a
b
b
a
b
a
2
111
tdt
dx
xt
tx
Example:
0
1
0
12
1 11)( dttgdt
ttfdxxf
0ab
If the integrand is singular at a known position c, than it is usually useful to split the integral as:
b
c
c
a
b
a
dxxfdxxfdxxf
)()()(
Note: Gaussian quadrature formulae never use the interval endpoints,which makes them very useful for the computation of improper integrals!