1000 CHAP. 24 Data Analysis. Probability Theory
18. Using a Venn diagram. show thatA g B if and only if (A°)° = A, 5" = Q, 6° = S,
AﬂB=A- AUA°=S, AnA°=Q
19. Show that, by the deﬁnition of complement, for any 20. Using a Venn diagram, show thatA ; B if and
SEC. 20.6 Matrix Eigenvalue Problems: Introduction 865
If we take equal factors together and denote the numerically distinct eigenvalues of A by
A1, - - - , A, (r E n), then the product becomes
(6) to) = Hm — Aomlo — A2)” - - ' (A — to"?
The exponent mj i
SEC. 7.3 Linear Systems of Equations. Gauss Elimination 289
E X A M P L E i Geometric Interpretation. Existence and Uniqueness of Solutions
If m = n = 2. we have two equations in two unknowns x1, x2
"ll-V1 + 012*}? = bl
a21x1 + (12212 = 172.
If we i
SEC. 18.5 Poisson’s Integral Formula for Potentials 769
We now write F(z) = (IJ(r, 9) + i‘l’(r, 0) and take the real part on both sides of (4).
Then we obtain Poisson’s integral formula2
2
11’ R2 _ .2
(MR, 0:) R ’ da.
1
(5) (130‘, 9) — — f m
27To
This for
SEC. 20.2 Linear Systems: LU-Factorization, Matrix Inversion
EXAMPLE 1
841
require fewer arithmetic operations. They are named after Doolittle, Crout, and Cholesky
and use the idea of the LU-factorization of A, which we explain first.
An LU-factorization
Summary of Chapter 16 727
_I._‘
.un‘-I
‘ -
LaUrent Series. Residue Integration
A Laurent series is a series of the form
so 30 bn
(I) .f(Z) = 2 an(z - 2:0)” + E —, (Sec. 16.1)
n=0 1L=1 (‘ — zo)
or, more brieﬂy written [but this means the same as (l)!]
SEC. 19.1
Introduction 781
Steps 1 and 2 are related. A slight change of the model may often admit of a more
efﬁcient method. To choose methods, we must ﬁrst get to know them. Chapters 19-21
contain efﬁcient algorithms for the most important classes of pr
SEC. 16.2 Singularities and Zeros. Infinity
THEOREM 1
EXAMPLE 3
THEOREM 2
EXAMPLE 4
709
Poles
If f(z) is analytic and has a pole at z = 20, then |f(z)| —> 00 as z -—> zo in any manner.
The proof is left to the student (see Prob. 12).
Behavior Near an
754
CHAP. 18 Complex Analysis and Potential Theory
Use of Conformal Mapping. Modeling
THEOREM 1
PROOF
EXAMPLE 1
Complex potentials relate potential theory closely to complex analysis, as we have just
seen. Another close relation results from the use of co
694
EXAMPLE 2
CHAP. 15 Power Series, Taylor Series
Series of Continuous Terms with a Discontinuous Sum
Consider the series
A2 x2 .t'2
+——+
1+.t-2 (I+.r2)2 (1
.t'2 + + ' - - (.t' real).
+ J?)3
This is a geometric series with q = l/(l + x2) times a factor
Numeric methods for differential equations are of great practical importance to the
engineer and physicist because practical problems often lead to differential equations that
cannot be solved by one of the methods in Chaps. 1—6 or 12 or by similar method
808 CHAP.19 Numerics in General
(b) Backward. For (18) we use j shown in the second column. and in each column the last number. Since
r = (L72 - 2.00)/0.l = —2.8. we thus gel from (18)
-2.8 - LS -2.8(- 1.8 -0.8
Joni/2) a 0.223 8908 - 2.8(—0.057 9278) + —-
850
1. Verify the claim at the end of Example 2.
2. Show that for the system in Example 2 the Jacobi
iteration diverges. Hint. Use eigenvalues.
CHAP. 2O Numeric Linear Algebra
Clearly, r = 0 if and only if x is a solution. Hence 1' at 0 for an approxima
946
Write in normal form and solve by the simplex method,
assuming all x,- to be nonnegative.
l. Maximize f = 3x, + 2x2 subject to 3x1 + 4x2 5 60,
4x1 + 3x2 g 60.
2. Prob. I6 in Problem Set 22.2.
3. Maximize the proﬁt in the daily production of x1 metal
f
976
THEOREM 1
PROOF
CHAP. 23 Graphs. Combinatorial Optimization
A “cut set” is a set of edges in a network. The underlying idea is simple and natural. If
we want to find out what is flowing from s to t in a network, we may cut the network
somewhere betwee
SEC. 19.5 Numeric Integration and Differentiation 817
Numeric Integration and Differentiation
Numeric integration means the numeric evaluation of integrals
b
J = I f(.\') dx
a
where a and b are given and f is a function given analytically by a formula or
958 CHAP. 23 Graphs. Combinatorial Optimization
1. Sketch the graph consisting of the vertices and edges
of a square. Of a tetrahedron.
2. Worker W, can do jobs .11 and 13, worker W2 job .14,
worker W3 jobs .12 and J3. Represent this by a graph.
3. Expl
874
EXAMPLE 1
EXAMPLE 2
CHAP. 20 Numeric Linear Algebra
Application of Theorem 1. Scaling
For the symmetric matrix A in Example 4, Sec. 20.7. and x0 = [I l I]T we obtain from (I) and (2) and the
indicated scaling
0.49 0.02 0.22 1 0.890244 0.931 193
A: 0.0
SEC. 21.5 Neumann and Mixed Problems. Irregular Boundary 919
Substituting these results into (2b) and simplifying. we have
21111 — 4ll12 + "22 = LS _ 3 = “1.5
21:21 + "12 — 4"” = 3 — 3 _ = ‘6.
Together with (221) this yields, written in matrix form.
-4 1
SEC. 21.1
EXAMPLE 5
Methods for First-Order ODEs
895
In both formulas we use only 6 different function evaluations altogether, namely,
k1 = hf(xﬂ3 yn)
k2 = l1f(.rn + ﬁll. )’n + %k1)
k3 = hf(.rn + 3/1. .Yn. + 5%](1 + '322'1‘2)
(14)
3 mm 72%
k4 = hf(xn + %1
904
EXAMPLE 2
CHAP. 21 Numerics for ODEs and PDEs
Table 21.1]
Euler Method for Systems in Example 1 (Mass—Spring System)
n x ‘11 y] Exact Error yz yz Exact Error
"‘ ' '" (5D) 61 = h * 3'1. ‘”’ (5D) 62 = .1’2 - Y2.
O 0.0 3.00000 3.00000 0.00000 —2.5000
SEC. 14.1 Line Integral in the Complex Plane 643
Solution. We may represent C in the form
2(r) = zD+p(cosr+ isini)= zo+peﬂ (ogigzw).
Then we have
(z _ Zorn = pmeimt’ d: = ipeit (I!
and obtain
21.- 217
§(, _ Zorn dz = I pmeimt ipez'l d). = iPm+1 I e‘i(m.+1
568 CHAP. 12 Partial Differential Equations (PDEs)
on the right. we obtain the solution fomiula
“DC
2 c2 2
(20) "(.r. I) = — f I f(p) sin wp e- w ‘ sin wx dp div.
7" o 0
Figure 297 shows (20) with c = I for f(.\') = I if 0 § x é I and 0 otherwise. graphed
SEC. 8.4 Eigenbases. Diagonalization. Quadratic Forms
20. CAS EXPERIMENT. Orthogonal Matrices.
(a) Products. Inverse. Prove that the product of two
orthogonal matrices is orthogonal. and so is the inverse
of an orthogonal matrix. What does this mean in te
SEC. 4.3 Constant~Coefficient Systems. Phase Plane Method 145
Find and graph 21 general solution of
I 4 l
(14) y = M = y
-l 2
Solution. A is not skew-symmetric! Its characteristic equation is
4—). l
=A2-6A+9=(A—3)2=0.
det(A-/\I)=|
-| 2-/\
It has a double
SEC. 5.5 Bessel’s Equation. BeSSel Functions 1.,(x)
THEOREM 1
THEOREM 2
193
Hence because of our (standard!) choice (18) of an the coefﬁcients (7) simply are
(19) _ (_ Um
azm 22"”"ml F(V + m + l) '
With these coefﬁcients and r = r1 = v we get from (2) a p
CHAPTER 22
CHAPTER 23
Unconstrained Optimization. Linear Programming
Graphs. Combinatorial Optimization
Ideas of optimization and application of graphs play an increasing role in engineering.
computer science, systems theory, economics. and other areas