Chow T.L. Mathematical Methods for Physicists: A Concise Introduction
Подождите немного. Документ загружается.
Equations of higher order. System of equations
The methods in the previous sections can be extended to obtain numerical solu-
tions of equations of higher order. An nth-order diÿerential equation is equivalent
to n ®rst-order diÿerential equations in n 1 variables. Thus, for instance, the
second-order equation
y
00
f x; y; y
0
; 13:27
with initial conditions
yx
0
y
0
; y
0
x
0
y
0
0
; 13:28
can be written as a system of two equations of ®rst order by setting
y
0
u; 13:29
then Eqs. (13.27) and (13.28) become
u
0
f x; y; u; 13:30
yx
0
y
0
; ux
0
u
0
: 13:31
The two ®rst-order equations (13.29) and (13.30) with the initial conditions
(13.31) are completely equivalent to the original second-order equation (13.27)
with the initial conditions (13.28). And the methods in the previous sections for
determining approximate solutions can be extended to solve this system of two
®rst-order equations. For example, the equation
y
00
ÿ y 2;
with initial conditions
y0ÿ1; y
0
01;
is equivalent to the system
y
0
x u; u
0
1 y;
with
y0ÿ1; u01:
These two ®rst-order equations can be solved with Taylor's method (Problem
13.12).
The simple methods outlined above all have the disadvantage that the error in
approximating to values of y is to a certain extent cumulative and may become
large unless some form of checking process is included. For this reason, methods
of solution involving ®nite diÿerence are devised, most of them being variations of
the Adams±Bashforth method that contains a self-checking process. This method
476
NUMERICAL METHODS
is quite involved and because of limited space we shall not cover it here, but it is
discussed in any standard textbook on numerical analysis.
Least-squares ®t
We now look at the problem of ®tting of experimental data. In some experimental
situations there may be underlying theory that suggests the kind of function to be
used in ®tting the data. Often there may be no theory on which to rely in selecting
a function to represent the data. In such circumstances a polynomial is often used.
We saw earlier that the m 1 coecients in the polynomial
y a
0
a
1
x a
m
x
m
can always be determined so that a given set of m 1 points (x
i
; y
i
), where the xs
may be unequal, lies on the curve described by the polynomial. However, when
the number of points is large, the degree m of the polynomial is high, and an
attempt to ®t the data by using a polynomial is very laborious. Furthermore, the
experimental data may contain experimental errors, and so it may be more
sensible to represent the data approximately by some function y f x that
contains a few unknown parameters. These parameters can then be determined
so that the curve y f x ®ts the data. How do we determine these unknown
parameters?
Let us represent a set of experimental data (x
i
; y
i
), where i 1; 2; ...; n, by some
function y f x that contains r parameters a
1
; a
2
; ...; a
r
. We then take the
deviations (or residuals)
d
i
f x
i
ÿy
i
13:32
and form the weighted sum of squares of the deviations
S
X
n
i1
w
i
d
i
2
X
n
i1
w
i
f x
i
ÿy
i
2
; 13:33
where the weights w
i
express our con®de nce in the accuracy of the experimental
data. If the points are equally weighted, the ws can all be set to 1.
It is clear that the quantity S is a function of as: S Sa
1
; a
2
; ...; a
r
: We can
now determine these parameters so that S is a minimum:
@S
@a
1
0;
@S
@a
2
0; ...;
@S
@a
r
0: 13: 34
The set of r equations (13.34) is called the normal equations and serves to
determine the r unknown asiny f x. This particular method of determining
the unknown as is known as the method of least squares.
477
LEAST-SQUARES FIT
We now illustrate the construction of the normal equations with the simplest
case: y f x is a linear function:
y a
1
a
2
x: 13:35
The deviations d
i
are given by
d
i
a
1
a
2
xÿy
i
and so, assuming w
i
1
S
X
n
i1
d
2
i
a
1
a
2
x
1
ÿ y
1
2
a
1
a
2
x
2
ÿ y
2
2
a
1
a
2
x
r
ÿ y
r
2
:
We now ®nd the partial derivatives of S with respect to a
1
and a
2
and set these to
zero:
@S=@a
1
2a
1
a
2
x
1
ÿ y
1
2a
1
a
2
x
2
ÿ y
2
2a
1
a
2
x
n
ÿ y
n
0;
@S=@a
2
2x
1
a
1
a
2
x
1
ÿ y
1
2x
2
a
1
a
2
x
2
ÿ y
2
2x
n
a
1
a
2
x
n
ÿ y
n
0:
Dividing out the fact or 2 and collecting the coecients of a
1
and a
2
, we obtain
na
1
X
n
i1
x
i
ÿ!
a
2
X
n
i1
y
1
; 13:36
X
n
i1
x
i
ÿ!
a
1
X
n
i1
x
2
i
ÿ!
a
2
X
n
i1
x
i
y
i
: 13:37
These equa tions can be solved for a
1
and a
2
.
Problems
13.1. Given six points ÿ1; 0, ÿ0:8; 2, ÿ0:6 ; 1, ÿ0:4 ; ÿ1, ÿ0:2; 0; and
0; ÿ4, determine a smooth function y f x such that y
i
f x
i
:
13.2. Find an approximate value of the real root of
x ÿ tan x 0
near x 3=2:
13.3. Find the angle subtended at the center of a circle by an arc whose length is
double the length of the chord.
13.4. Use Newton's method to solve
e
x
2
ÿ x
3
3x ÿ 4 0;
with x
0
0 and h 0:001:
478
NUMERICAL METHODS
13.5. Use Newton's method to ®nd a solution of
sinx
3
21=x;
with x
0
1andh 0:001.
13.6. Approximate the following integrals using the rectangular rule, the
trapezoidal rule, and Simpson's rule, with n 2; 4; 10; 20; 50:
(a)
Z
=2
0
e
ÿx
2
sinx
2
1 dx;
(b)
Z
2
p
0
sinx
2
3x ÿ 2
x 4
dx;
(c)
Z
1
0
dx
2 ÿ sin
2
x
p
:
13.7 Show that the area under a parabola, as shown in Fig. 13.7, is given by
A
h
3
y
1
4y
2
y
3
:
13.8. Using the improved Euler's method, ®nd the value of y when x 0:2on
the integral curve of the equation y
0
x
2
ÿ 2y through the point x 0,
y 1.
13.9. Using Taylor's method, ®nd correct to four places of decimals values of y
corresponding to x 0:2 and x ÿ0:2 for the solution of the diÿerential
equation
dy=dx x ÿ y
2
=10;
with the initial condition y 1 when x 0.
13.10. Using the Runge±Kutta method and h 0:1, solve
y
0
x
2
ÿ siny
2
; x
0
1 and y
0
4:7:
13.11. Using the Runge±Kutta method and h 0:1, solve
y
0
ye
ÿx
2
; x
0
1 and y
0
3:
13.12. Using Taylor's method, obtain the solution of the system
y
0
x u; u
0
1 y
with
y0ÿ1; u01:
.
13.13. Find to four places of decimals the solution between x 0andx 0:5of
the equati ons
y
0
1
2
y u ; u
0
1
2
y
2
ÿ u
2
;
with y u 1 when x 0.
479
PROBLEMS
13.14. Find to three places of decimals a solution of the equation
y
00
2xy
0
ÿ 4y 0;
with y y
0
1 when x 0:
13.15. Use Eqs. (13.36) and (13.37) to calcul ate the coecients in y a
1
a
2
x to
®t the following data: x; y1; 1:7; 2; 1: 8; 3; 2:3; 4; 3:2:
480
NUMERICAL METHODS
14
Introduction to probability theory
The theory of probability is so useful that it is required in almost every branch of
science. In physics, it is of basic importance in quantum mechanics, kinetic theory,
and thermal and statistical physics to name just a few topics. In this chapter the
reader is introduced to some of the fundamental ideas that make probability
theory so useful. We begin with a review of the de®nitions of probability, a
brief discussion of the fundamental laws of probability, and methods of counting
(some facts about permutations and combinations), probability distributions are
then treated.
A notion that will be used very often in our discussion is `equally likely'. This
cannot be de®ned in terms of anything simpler, but can be explained and illu-
strated with simple examples. For example, heads and tails are equally likely
results in a spin of a fair coin; the ace of spades and the ace of hearts are equally
likely to be drawn from a shued deck of 52 cards. Many more examples can be
given to illustrate the concept of `equally likely'.
A de®nition of probability
Now a question that arises naturally is that of how shall we measure the
probability that a particular case (or outcome) in an experiment (such as the
throw of dice or the draw of cards) out of many equall y likely cases that will
occur. Let us ¯ip a coin twice, and ask the question: what is the probability of it
coming down heads at least once. There are four equally likely results in ¯ipping a
coin twice: HH, HT, TH, TT, where H stands for head and T for tail. Three of the
four results are favorable to at least one head showing, so the probability of
getting one head is 3/4. In the example of drawn cards, what is the probability
of drawing the ace of spades? Obviously there is one chance out of 52, and the
probability, accordingly, is 1/52. On the other hand, the probability of drawing an
481
ace is four times as great ÿ4=52, for there are four aces, equally likely. Reasoning
in this way, we are led to give the notion of probability the following de®nition:
If there are N mutually exclusive, collective exhaustive, and
equally likely outcomes of an experiment, and n of these are
favorable to an event A, then the probability pA of an event
A is n=N: p n=N,or
pA
number of outcomes favorable to A
total number of results
: 14:1
We have made no attempt to predict the result, just to measure it. The de®nition
of probability given here is often called a posteriori probability.
The terms exclus ive and exhaustive ne ed some attention. Two events are said to
be mutually exclusive if they cannot both occur together in a single trial; and the
term collective exhaustive means that all possible outcomes or results are enum-
erated in the N outcomes.
If an event is certain not to occur its probability is zero, and if an event is
certain to occur, then its probability is 1. Now if p is the probability that an eve nt
will occur, then the probability that it will fail to occur is 1 ÿ p, and we denote it
by q:
q 1 ÿ p: 14:2
If p is the probability that an event will occur in an experiment, and if the
experiment is repeated M times, then the expected number of times the event will
occur is Mp. For suciently large M, Mp is expected to be close to the actual
number of times the event will occur. For example, the probability of a head
appearing when tossing a coin is 1/2, the expected number of times heads appear
is 4 1=2 or 2. Actually, heads will not always appear twice when a coin is tossed
four times. But if it is tossed 50 times, the number of heads that appear will, on the
average, be close to 25 50 1=2 25). Note that closeness is computed on a
percentage basis: 20 is 20% of 25 away from 25 while 1 is 50% of 2 away from 2.
Sample space
The equally likely cases associated with an experiment represent the possible out-
comes. For example, the 36 equally likel y cases associated with the throw of a pair
of dice are the 36 ways the dice may fall, and if 3 coins are tossed, there are 8
equally likely cases corresponding to the 8 possible outcomes. A list or set that
consists of all possible outcomes of an experiment is called a sample space and
each individual outcome is called a sample point (a point of the sample space).
The outcomes composing the sample space are req uired to be mutually exclusive.
As an example, when tossing a die the outcomes `an even number shows' and
482
INTRODUCTION TO PROBABILITY THEORY
`number 4 shows' cannot be in the same sample space. Often there will be more
than one sample space that can describe the outcome of an experiment but there is
usually only one that will provide the most information. In a throw of a fair die,
one sample space is the set of all possible outcomes {1, 2, 3, 4, 5, 6}, and another
could be {even} or {odd}.
A ®nite sample space is one that has only a ®nite number of points. The points
of the sample space are weighted according to their prob abilities. To see this, let
the points have the probabilities
p
1
; p
2
; ...; p
N
with
p
1
p
2
p
N
1:
Suppose the ®rst n sample points are favorable to another event A. Then the
probability of A is de®ned to be
pAp
1
p
2
p
n
:
Thus the points of the sample space are weighted according to their probabilities.
If each point has the sampl e probability 1/n, then pA becomes
pA
1
N
1
N
1
N
n
N
and this de®nition is consistent with that given by Eq. (14.1).
A sample space with constant probab ility is called uniform. Non-uniform sam-
ple spaces are more common. As an example, let us toss four coins and count the
number of heads. An appropriate sample space is composed of the outcomes
0 heads; 1 head; 2 heads; 3 heads; 4 heads;
with respective probabilities, or weights
1=16; 4=16; 6=16; 4=16; 1=16:
The four coins can fall in 2 2 2 2 2
4
, or 16 ways. They give no heads (all
land tails) in only one outcome, and hence the required probability is 1/16. There
are four ways to obtain 1 head: a head on the ®rst coin or on the second coin, and
so on. This gives 4/16. Similarly we can obtain the probabilities for the other
cases.
We can also use this simple example to illustrate the use of sample space. What
is the pro bability of getting at least two heads? Note that the last three sample
points are favorable to this event, hence the required probability is given by
6
16
4
16
1
16
11
16
:
483
SAMPLE SPACE
Methods of counting
In many applications the total number of elements in a sample space or in an
event needs to be counted. A fundamental principle of counting is this: if one
thing can be done in n diÿerent ways and another thing can be done in m diÿerent
ways, then both things can be done together or in succession in mn diÿerent ways.
As an example, in the example of throwing a pair of dice cited above, there are 36
equally like outcomes: the ®rst die can fall in six ways, and for each of these the
second die can also fall in six ways. The total number of ways is
6 6 6 6 6 6 6 6 36
and these are equally likely.
Enumeration of outcomes can become a lengthy process, or it can become a
practical impossibility. For example, the throw of four dice generates a sample
space with 6
4
1296 elements. Some systematic methods for the counting are
desirable. Permutation and combination formulas are often very useful.
Permutations
A permutation is a particular ordered selection. Suppose there are n objects and r
of these objects are arranged into r numbered spaces. Since there are n ways of
choosing the ®rst object, and after this is done there are n ÿ 1 ways of choo sing
the second object, ...; and ®nally n ÿr ÿ 1 ways of choosing the rth object, it
follows by the fundamental principle of counting that the number of diÿerent
arrangements or permutations is given by
n
P
r
nn ÿ 1 n ÿ 2n ÿ r 1: 14:3
where the product on the right-hand side has r factors. We call
n
P
r
the number of
permutations of n objects taken r at a time. When r n, we have
n
P
n
nn ÿ 1 n ÿ 21 n!:
We can rewrite
n
P
r
in terms of factorials:
n
P
r
nn ÿ 1 n ÿ 2n ÿ r 1
nn ÿ 1 n ÿ 2n ÿ r 1
n ÿ r 2 1
n ÿ r 2 1
n!
n ÿ r !
:
When r n,wehave
n
P
n
n!=n ÿ n! n!=0!. This reduces to n! if we have
0! 1 and mathematicians actually take this as the de®nition of 0!.
Suppose the n objects are not all diÿerent . Instead, there are n
1
objects of one
kind (that is, indistinguishable from each other), n
2
that is of a second kind; ...; n
k
484
INTRODUCTION TO PROBABILITY THEORY
of a kth kind so that n
1
n
2
n
k
n. A natural question is that of how
many distinguishable arrangements are there of these n objects. Assuming that
there are N diÿerent arrangements, and each distinguishable arrangement appears
n
1
!, n
2
!; ... times, where n
1
! is the number of ways of arranging the n
1
objects,
similarly for n
2
!; ...; n
k
!: Then multiplying N by n
1
!n
2
!; ...; n
k
! we obtain the
number of ways of arranging the n objects if they were all distinguishable, that
is,
n
P
n
n!:
Nn
1
!n
2
! n
k
! n! or N n!=n
1
!n
2
!...n
k
!:
N is often written as
n
P
n
1
n
2
:::n
k
, and then we have
n
P
n
1
n
2
:::n
k
n!
n
1
!n
2
! n
k
!
: 14:4
For example, given six coins: one penny, two nickels and three dimes, the number
of permutations of these six coins is
6
P
123
6!=1!2!3! 60:
Combinations
A permutation is a particular ordered selection. Thus 123 is a diÿerent permuta-
tion from 231. In many problems we are interested only in selecting objects with-
out regard to order. Such selections are called combinations. Thus 123 and 231
are now the same combination. The notation for a combination is
n
C
r
which
means the number of ways in which r objects can be selected from n objects
without regard to order (also called the combination of n objects taken r at a
time). Among the
n
P
r
permutations there are r! that give the same combinat ion.
Thus, the total number of permutations of n diÿerent objects selected r at a time is
r!
n
C
r
n
P
r
n!
n ÿ r !
:
Hence, it follows that
n
C
r
n!
r!n ÿ r!
: 14:5
It is straightforward to show that
n
C
r
n!
r!n ÿ r!
n!
n ÿn ÿ r!n ÿ r!
n
C
nÿr
:
n
C
r
is often written as
n
C
r
n
r
:
485
METHODS OF COUNTING