# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

6 Pages

### Computer Arithmetic Notes

Course: MATH 5485, Fall 2009
School: UCF
Rating:

Word Count: 2150

#### Document Preview

Lecture AIMS Notes 2006 Peter J. Olver 1. Computer Arithmetic The purpose of computing is insight, not numbers. -- R.W. Hamming, [23] The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathematical quantities, including functions, integrals, solutions of algebraic equations, solutions of differential equations (both ordinary and partial), solutions of...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Florida >> UCF >> MATH 5485

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Lecture AIMS Notes 2006 Peter J. Olver 1. Computer Arithmetic The purpose of computing is insight, not numbers. -- R.W. Hamming, [23] The main goal of numerical analysis is to develop efficient algorithms for computing precise numerical values of mathematical quantities, including functions, integrals, solutions of algebraic equations, solutions of differential equations (both ordinary and partial), solutions of minimization problems, and so on. The objects of interest typically (but not exclusively) arise in applications, which seek not only their qualitative properties, but also quantitative numerical data. The goal of this course of lectures is to introduce some of the most important and basic numerical algorithms that are used in practical computations. Beyond merely learning the basic techniques, it is crucial that an informed practitioner develop a thorough understanding of how the algorithms are constructed, why they work, and what their limitations are. In any applied numerical computation, there are four key sources of error: (i ) Inexactness of the mathematical model for the underlying physical phenomenon. (ii ) Errors in measurements of parameters entering the model. (iii ) Round-off errors in computer arithmetic. (iv ) Approximations used to solve the full mathematical system. Of these, (i ) is the domain of mathematical modeling, and will not concern us here. Neither will (ii ), which is the domain of the experimentalists. (iii ) arises due to the finite numerical precision imposed by the computer. (iv ) is the true domain of numerical analysis, and refers to the fact that most systems of equations are too complicated to solve explicitly, or, even in cases when an analytic solution formula is known, directly obtaining the precise numerical values may be difficult. There are two principal ways of quantifying computational errors. Definition 1.1. Let x be a real number and let x be an approximation. The absolute error in the approximation x x is defined as | x - x |. The relative error is |x - x| , which assumes defined as the ratio of the absolute error to the size of x, i.e., |x| x = 0; otherwise relative error is not defined. 3/15/06 1 c 2006 Peter J. Olver For example, 1000001 is an approximation to 1000000 with an absolute error of 1 and a relative error of 10-6 , while 2 is an approximation to 1 with an absolute error of 1 and a relative error of 1. Typically, relative error is more intuitive and the preferred determiner of the size of the error. The present convention is that errors are always 0, and are = 0 if and only if the approximation is exact. We will say that an approximation x has k significant decimal digits if its relative error is < 5 10-k-1 . This means that the first k digits of x following its first nonzero digit are the same as those of x. Ultimately, numerical algorithms must be performed on a computer. In the old days, "computer" referred to a person or an analog machine, but nowadays inevitably means a digital, electronic machine. A most unfortunate fact of life is that all digital computers, no matter how "super", can only store finitely many quantities. Thus, there is no way that a computer can represent the (discrete) infinity of all integers or all rational numbers, let alone the (continuous) infinity of all real or all complex numbers. So the decision of how to approximate more general numbers using only the finitely many that the computer can handle becomes of critical importance. Each number in a computer is assigned a location or word , consisting of a specified number of binary digits or bits. A k bit word can store a total of N = 2 k different numbers. For example, the standard single precision computer uses 32 bit arithmetic, for a total of N = 232 4.3 109 different numbers, while double precision uses 64 bits, with N = 264 1.84 1019 different numbers. The question is how to distribute the N exactly representable numbers over the real line for maximum efficiency and accuracy in practical computations. One evident choice is to distribute them evenly, leading to fixed point arithmetic. For example, one can use the first bit in a word to represent a sign, and treat the remaining bits as integers, thereby representing the integers from 1 - 1 N = 1 - 2k-1 to 1 N = 2k-1 2 2 exactly. Of course, this does a poor job approximating most non-integral numbers. Another option is to space the numbers closely together, say with uniform gap of 2 -n , and so distribute our N numbers uniformly over the interval - 2- n-1 N < x 2- n-1 N . Real numbers lying between the gaps are represented by either rounding, meaning the closest exact representative, or chopping, meaning the exact representative immediately below (or above if negative) the number. Numbers lying beyond the range must be represented by the largest (or largest negative) representable number, which thus becomes a symbol for overflow. When processing speed is a significant bottleneck, the use of such fixed point representations is an attractive and faster alternative to the more cumbersome floating point arithmetic most commonly used in practice. The most common non-uniform distribution of our N numbers is the floating point system, which is based on scientific notation. If x is any real number it can be written in the form x = .d1 d2 d3 d4 . . . 2n , where d = 0 or 1, and n Z is an integer. We call d1 d2 d3 d4 . . . the mantissa and n the exponent. If x = 0, then we can uniquely choose n so that d1 = 1. In our computer, we approximate x by a finite number of mantissa digits x = .d1 d2 d3 d4 . . . dk-1 dk 2n , 3/15/06 2 c 2006 Peter J. Olver by either chopping or rounding the final digit. The exponent is also restricted to a finite range of integers n N N . Very small numbers, lying in the gap between 0 < | x | < 2n , are said to cause underflow . In single precision floating point arithmetic, the sign is 1 bit, the exponent is 7 bits, and the mantissa is 24 bits. The resulting nonzero numbers lie in the range 2-127 10-38 | x | 2127 1038 , and allow one to accurately represent numbers with approximately 7 significant decimal digits of real numbers lying in this range. In double precision floating point arithmetic, the sign is 1 bit, the exponent is 10 bits, and the mantissa is 53 bits, leading to floating point representations for a total of 1.84 1019 different numbers which, apart from 0. The resulting nonzero numbers lie in the range 2-1023 10-308 | x | 21023 10308 . In double precision, one can accurately represent approximately 15 decimal digits. Keep in mind floating point numbers are not uniformly spaced ! Moreover, when passing from .111111 . . . 2n to .100000 . . . 2n+1 , the inter-number spacing suddenly jumps by a factor of 2. The non-smoothly varying spacing inherent in the floating point system can cause subtle, undesirable numerical in artifacts high precision computations. Remark : Although they are overwhelmingly the most prevalent, fixed and floating point are not the only number systems that have been proposed. See [9] for another intriguing possibility. In the course of a calculation, intermediate errors interact in a complex fashion, and result in a final total error that is not just the sum of the individual errors. If x is an approximation to x, and y is an approximation to y, then, instead of arithmetic operations +, -, , / and so on, the computer uses a "pseudo-operations" to combine them. For instance, to approximate the difference x - y of two real numbers, the computer begins by replacing each by its floating point approximation x and y . It the subtracts these, and replaces the exact result x - y by its nearest floating point representative, namely (x - y ) . As the following example makes clear, this is not necessarily the same as the floating point approximation to x - y, i.e., in general (x - y ) = (x - y) . Example 1.2. . Lets see what happens if we subtract the rational numbers 301 301 .15050000 . . . , y= .150424787 . . . . 2000 2001 The exact answer is 301 x-y = .00007521239 . . . . 4002000 However, if we use rounding to approximate with 4 significant digits , then x= x= 301 .1505, 2000 y= 301 .1504 2001 and so x - y .0001, To aid comprehension, we are using base 10 instead of base 2 arithmetic in this example. 3/15/06 3 c 2006 Peter J. Olver 3 2 1 2 -1 -2 -3 4 6 8 10 Figure 1.1. Roots of Polynomials. which has no significant digits in common with the actual answer. On the other hand, if we evaluate the difference using the alternative formula x-y = 301 2001 - 301 2000 602301 - 602000 = 4002000 4002000 5 5 6.023 10 - 6.020 10 .003 105 = .00007496, 4.002 106 4.002 106 we at least reproduce the first two significant digits. Thus, the result of a floating point computation depends on the order of the arithmetic operations! In particular, the associative and distributive laws are not valid in floating point arithmetic! In the development of numerical analysis, one tends to not pay attention to this "minor detail', although its effects must always be kept in the back of one's mind when evaluating the results of any serious numerical computation. Just in case you are getting a little complacent, thinking "gee, a few tiny round off errors can't really make that big a difference", let us close with two cautionary examples. Example 1.3. Consider the pair of degree 10 polynomials p(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(x - 8)(x - 9)(x - 10) and q(x) = p(x) + x5 . They only differ in the value of the coefficient of their middle term, which, by a direct expansion, is -902055 x5 in p(x), and - 902054 x5 in q(x); all other terms are the same. The relative error between the two coefficients is roughly one-thousandth of one percent. Our task is to compute the roots of these two polynomials, i.e., the solutions to p(x) = 0 and q(x) = 0. Those of p(x) are obvious. One might expect the roots of q(x) to 3/15/06 4 c 2006 Peter J. Olver 1 0.8 0.6 0.4 0.2 0.5 1 1.5 2 2.5 Figure 1.2. Sensitive Dependence on Initial Data. be rather close to the integers 1, 2, . . . , 10. However, their approximate values are 1.0000027558, 5.24676 0.751485 i , 1.99921, 7.57271 1.11728 i , 3.02591, 9.75659 0.368389 i . 3.82275, Surprisingly, only the smallest two roots are relatively unchanged; the third and fourth roots differ by roughly 1% and 27%, while the final six roots have mysteriously metamorphosed into three complex conjugate pairs of roots of q(x). The two sets of roots are plotted in Figure 1.1; those of p(x) are indicated by solid disks, while those of q(x) are indicated by open circles. We have thus learned that an almost negligible change in a single coefficient of a real polynomial can have dramatic and unanticipated effects on its roots. Needless to say, this indicates that finding accurate numerical values for the roots of high degree polynomials is a very challenging problem. Example 1.4. Consider the initial value problem du - 2 u = - e- t , u(0) = 1 . 3 dt The solution is easily found: u(t) = 1 e- t , 3 and is exponentially decaying as t . However, in a floating point computer, we are not able to represent the initial con1 dition 3 exactly, and make some small round-off error (depending on the precision of the computer). Let = 0 represent the error in the initial condition. The solution to the perturbed initial value problem dv - 2 v = - e- t , dt is v(t) = 1 -t 3e v(0) = + e2 t , 1 3 + , which is exponentially growing as t increases. As sketched in Figure 1.2, the two solutions remain close only for a very short time interval, the duration of which depends on the 3/15/06 5 c 2006 Peter J. Olver size in the initial error, but then they eventually diverge far away from each other. As a consequence, a tiny error in the initial data can have a dramatic effect on the solution. This phenomenon is referred to as sensitive dependence on initial conditions. The numerical computation of the exponentially decaying exact solution in the face of round-off errors is an extreme challenge. Even the tiniest of error will immediately introduce an exponentially growing mode which will eventually swamp the true solution. Furthermore, in many important applications, the physically relevant solution is the one that decays to zero at large distances, and is usually distinguished from the vast majority of solutions that grow at large distances. So the computational problem is both important and very difficult. Examples 1.3 and 1.4 are known as ill-conditioned problems meaning that tiny changes in the data have dramatic effects on the solutions. Simple numerical methods work as advertised on well-conditioned problems, but all have their limitations and a sufficiently ill-conditioned problem will test the limits of the algorithm and/or computer, and, in many instances, require revamping a standard numerical solution scheme to adapt to the ill-conditioning. Some problems are so ill-conditioned as to defy even the most powerful computers and sophisticated algorithms. For this reason, numerical analysis will forever remain a vital and vibrant area of mathematical research. So, numerical analysis cannot be viewed in isolation as a black box, and left in the hands of the mathematical experts. Every practitioner will, sooner or later, confront a problem that tests the limitations of standard algorithms and software. Without a proper understanding of the mathematical principles involved in constructing basic numerical algorithms, one is ill-equipped, if not hopelessly handicapped, when dealing with such problems. The purpose of this series of lectures is to give you the proper mathematical grounding in modern numerical analysis. 3/15/06 6 c 2006 Peter J. Olver
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

UCF - MATH - 5485
Chapter 21 The Calculus of VariationsWe have already had ample opportunity to exploit Nature's propensity to minimize. Minimization principles form one of the most wide-ranging means of formulating mathematical models governing the equilibrium configurat
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver3. Review of Matrix AlgebraVectors and matrices are essential for modern analysis of systems of equations - algebrai, differential, functional, etc. In this part, we will review the most basic facts of matrix arithm
UCF - MATH - 5485
Chapter 13 Fourier AnalysisIn addition to their inestimable importance in mathematics and its applications, Fourier series also serve as the entry point into the wonderful world of Fourier analysis and its wide-ranging extensions and generalizations. An
UCF - MATH - 5485
Chapter 12 Fourier SeriesJust before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. As a result of his investigations into the partial differential equations modeling vibration and heat propa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver4. Gaussian EliminationIn this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor of one of the all-time mathematical greats - the early nin
UCF - MATH - 5485
Chapter 14 Vibration and Diffusion in OneDimensional MediaIn this chapter, we study the solutions, both analytical and numerical, to the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffusion of thermal
UCF - MATH - 5485
Math 5485 September 11, 2006Homework #1Problems: 1.3 1.4 2.1 1b, 3, 4a,b(single precision only), 9. 1a, 2, 7, 13. 1d, 3, 8, 11, 16a.Due: Wednesday, September 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 September 20, 2006Homework #2Problems: 2.2 2.3 2.4 1d, 5 (only do parts 1 &amp; 3), 13. 1, 5, 7, 11. 1d, 4, 9, 14a.Due: Wednesday, September 27 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 September 27, 2006Homework #3Problems: 2.5 2.6 1d, 6, 11a. 1, 5, 8.Due: Wednesday, October 4 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 October 4, 2006Homework #4Problems: 3.1 3.2 3.5 1, 8, 10, 12b. 7a, 14, 18b. 10a, 11.Due: Friday, October 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm: Wednesday, November 1 Will cover chapters 1, 2, 3. You
UCF - MATH - 5485
Math 5485 October 13, 2006Homework #5Problems: 3.3 3.6 3.7 2a, 3a, b(a), c, 5b, d (also, what is the spectral radius?), 6b, c, 7a, 10. 2,10. 14b, 19.Due: Friday, October 20 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm:
UCF - MATH - 5485
Math 5485 October 23, 2006Homework #6Problems: 3.7 3.8 5b, 6. 3a, 9, 11 (for 9), 12 (for 9), 13.Due: Monday, November 6 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.First Midterm: Wednesday, November 1 Will cover sections 1.24, 2.16,
UCF - MATH - 5485
Math 5485 November 15, 2006Homework #7Problems: 3.10 4.1 4.2 4.3 7 (just do Newton's Method), 11b. 2, 11, 14a, 15a. 2, 8, 10. 1b, 5.Due: Monday, November 27 Text: B. Bradie, A Friendly Introduction to Numerical Analysis.
UCF - MATH - 5485
Math 5485 November 27, 2006Homework #8Problems: 4.4 4.5 1ac, 4a, 5b, 8. 6 (ignore the Wilkinson shift), 12 (compare the convergence rate of the direct QR algorithm with that based on tridiagonalization).Due: Monday, December 4 Text: B. Bradie, A Friend
UCF - MATH - 5485
Math 5485 December 4, 2006Homework #9Problems: 5.1 5.2 5.3 5.4 2, 5, 8. 1b, 4, 9, 12. 4, 8, 11. 2, 3, 10 (only uniform and Chebyshev).Due: Wednesday, December 13 Text: B. Bradie, A Friendly Introduction to Numerical Analysis. Second Midterm: Friday, De
UCF - MATH - 5485
Chapter 17 Dynamics of Planar MediaIn this chapter, we continue our ascent of the dimensional ladder for linear systems. In Chapter 6, we embarked on our journey with equilibrium configurations of discrete systems - massspring chains, circuits, and struc
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver7. Iterative Methods for Linear SystemsLinear iteration coincides with multiplication by successive powers of a matrix; convergence of the iterates depends on the magnitude of its eigenvalues. We discuss in some det
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver5. Inner Products and NormsThe norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number of other important norms that are used in numerical analysis
UCF - MATH - 5485
Chapter 15 The Planar Laplace EquationThe fundamental partial differential equations that govern the equilibrium mechanics of multi-dimensional media are the Laplace equation and its inhomogeneous counterpart, the Poisson equation. The Laplace equation i
UCF - MATH - 5485
Very Basic MATLABPeter J. Olver January, 2009 Matrices: Type your matrix as follows: Use space or , to separate entries, and ; or return after each row. &gt; A = [4 5 6 -9;5 0 -3 6;7 8 5 0; -1 4 5 1] or &gt; A = [4,5,6,-9;5,0,-3,6;7,8,5,0;-1,4,5,1] or &gt; A = [
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver9. Numerical Solution of Algebraic SystemsIn this part, we discuss basic iterative methods for solving systems of algebraic equations. By far the most common is a vector-valued version of Newton's Method, which will
UCF - MATH - 5485
Chapter 19 Nonlinear SystemsNonlinearity is ubiquitous in physical phenomena. Fluid and plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inhe
UCF - MATH - 5485
Chapter 22 Nonlinear Partial Differential EquationsThe ultimate topic to be touched on in this book is the vast and active field of nonlinear partial differential equations. Leaving aside quantum mechanics, which remains to date an inherently linear theo
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver11. Numerical Solution of the Heat and Wave EquationsIn this part, we study numerical solution methodss for the two most important equations of one-dimensional continuum dynamics. The heat equation models the diffus
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver10. Numerical Solution of Ordinary Differential EquationsThis part is concerned with the numerical solution of initial value problems for systems of ordinary differential equations. We will introduce the most basic
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver8. Numerical Computation of EigenvaluesIn this part, we discuss some practical methods for computing eigenvalues and eigenvectors of matrices. Needless to say, we completely avoid trying to solve (or even write down
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver13. Approximation and InterpolationWe will now apply our minimization results to the interpolation and least squares fitting of data and functions.13.1. Least Squares.Linear systems with more equations than unknow
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver2. Numerical Solution of Scalar EquationsMost numerical solution methods are based on some form of iteration. The basic idea is that repeated application of the algorithm will produce closer and closer approximation
UCF - MATH - 5485
Chapter 20 Nonlinear Ordinary Differential EquationsThis chapter is concerned with initial value problems for systems of ordinary differential equations. We have already dealt with the linear case in Chapter 9, and so here our emphasis will be on nonline
UCF - MATH - 5485
Chapter 18 Partial Differential Equations in ThreeDimensional SpaceAt last we have ascended the dimensional ladder to its ultimate rung (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. A
UCF - MATH - 5485
Orthogonal Bases and the QR Algorithmby Peter J. Olver University of Minnesota1. Orthogonal Bases.Throughout, we work in the Euclidean vector space V = R n , the space of column vectors with n real entries. As inner product, we will only use the dot pr
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver14. Finite ElementsIn this part, we introduce the powerful finite element method for finding numerical approximations to the solutions to boundary value problems involving both ordinary and partial differential equa
UCF - MATH - 5485
AIMS Lecture Notes 2006Peter J. Olver12. MinimizationIn this part, we will introduce and solve the most basic mathematical optimization problem: minimize a quadratic function depending on several variables. This will require a short introduction to pos
UCF - MATH - 5587
Remark : On a connected domain R 2 , all harmonic conjugates to a given function u(x, y) only differ by a constant: v(x, y) = v(x, y) + c; see Exercise . Although most harmonic functions have harmonic conjugates, unfortunately this is not always the case.
UCF - MATH - 5587
Chapter 7 Complex Analysis and Conformal MappingThe term &quot;complex analysis&quot; refers to the calculus of complex-valued functions f (z) depending on a single complex variable z. To the novice, it may seem that this subject should merely be a simple reworkin
UCF - MATH - 5587
1 Re z Figure 7.1.1 Im z 1 Real and Imaginary Parts of f (z) = z .Therefore, if f (z) is any complex function, we can write it as a complex combination f (z) = f (x + i y) = u(x, y) + i v(x, y), of two inter-related real harmonic functions: u(x, y) = Re
UCF - MATH - 5587
Figure 7.4.Real and Imaginary Parts ofz.also have complex logarithms! On the other hand, if z = x &lt; 0 is real and negative, then log z = log | x | + (2 k + 1) i is complex no matter which value of ph z is chosen. (This explains why one avoids defining
UCF - MATH - 5587
The proof of the converse - that any function whose real and imaginary components satisfy the CauchyRiemann equations is differentiable - will be omitted, but can be found in any basic text on complex analysis, e.g., [3, 65, 118]. Remark : It is worth poi
UCF - MATH - 5587
is analytic everywhere except for singularities at the points z = 3 and z = -1, where its denominator vanishes. Since f (z) = h1 (z) , z-3 where h1 (z) = ez (z + 1)21 is analytic at z = 3 and h1 (3) = 16 e3 = 0, we conclude that z = 3 is a simple (order
UCF - MATH - 5587
if and only if it has vanishing divergence: v = u v + = 0. x y (7.36)Incompressibility means that the fluid volume does not change as it flows. Most liquids, including water, are, for all practical purposes, incompressible. On the other hand, the flow is
UCF - MATH - 5587
Using formula (7.19) for the complex derivative, d = -i = u - i v, dz x y so = u, x = v. yThus, = v, and hence the real part (x, y) of the complex function (z) defines a velocity potential for the fluid flow. For this reason, the anti-derivative (z) is k
UCF - MATH - 5587
gDFigure 7.14.Mapping to the Unit Disk.Remark : In this section, we have focused on the fluid mechanical roles of a harmonic function and its conjugate. An analogous interpretation applies when (x, y) represents an electromagnetic potential function;
UCF - MATH - 5587
Figure 7.16.The Effect of = z 2 on Various Domains.obtained by cutting the complex plane along the negative real axis. On the other hand, vertical lines Re z = a are mapped to circles | | = ea . Thus, a vertical strip a &lt; Re z &lt; b is mapped to an annulu
UCF - MATH - 5587
zph zFigure 7.18.Complex Curve and Tangent.notation x(t) = ( x(t), y(t) ) to complex notation z(t) = x(t)+ i y(t). All the usual vectorial curve terminology - closed, simple (non-self intersecting), piecewise smooth, etc. - is employed without modific
UCF - MATH - 5587
Center: .1 Radius: .5Center: .2 + i Radius: 1Center: 1 + i Radius: 1Center: -2 + 3 i Radius: 3 2 4.2426Center: .2 + i Radius: 1.2806 Figure 7.21.Center: .1 + .3 i Radius: .9487Center: .1 + .1 i Radius: 1.1045Center: -.2 + .1 i Radius: 1.2042Airfoi
UCF - MATH - 5587
Example 7.35. The goal of this example is to construct an conformal map that takes a half disk D+ = | z | &lt; 1, Im z &gt; 0 (7.73) to the full unit disk D = cfw_ | | &lt; 1 . The answer is not = z 2 because the image of D+ omits the positive real axis, resulting
UCF - MATH - 5587
7.5. Applications of Conformal Mapping.Let us now apply what we have learned about analytic/conformal maps. We begin with boundary value problems for the Laplace equation, and then present some applications in fluid mechanics. We conclude by discussing h
UCF - MATH - 5587
Figure 7.25.A NonCoaxial Cable.Example 7.39. A non-coaxial cable. The goal of this example is to determine the electrostatic potential inside a non-coaxial cylindrical cable, as illustrated in Figure 7.25, with prescribed constant potential values on th
UCF - MATH - 5587
0 Figure 7.29.15 Fluid Flow Past a Tilted Plate.30Note that = ( 1, 0 ), and hence this flow satisfies the Neumann boundary conditions (7.95) on the horizontal segment D = . The corresponding complex potential is (z) = z, with complex velocity f (z) = (
UCF - MATH - 5587
on the unit disk D for an impulse concentrated at the origin; see Section 6.3 for details. How do we obtain the corresponding solution when the unit impulse is concentrated at another point = + i D instead of the origin? According to Example 7.25, the lin
UCF - MATH - 5587
as long as n = -1. Therefore, we can use the Fundamental Theorem of Calculus (which works equally well for real integrals of complex-valued functions), to evaluate n+1 1 -1 = n = 2 k + 1 odd, 0, 2 t + i (t - 1) 2 z n dz = = , n = 2 k even. n+1 P t = -1 n+
UCF - MATH - 5587
Figure 7.32.Orientation of Domain Boundary.Theorem 7.48. If f (z) is analytic on a bounded domain C, then f (z) dz = 0.(7.118)Proof : If we apply Green's Theorem to the two real line integrals in (7.109), we find u dx - v dy = - u v - x y = 0,v dx +
UCF - MATH - 5587
Proof : Note that the integrand f (z) = 1/(z - a) is analytic everywhere except at z = a, where it has a simple pole. If a is outside C, then Cauchy's Theorem 7.48 applies, and the integral is zero. On the other hand, if a is inside C, then Proposition 7.
UCF - MATH - 5587
0 Figure 7.36.15 Kutta Flow Past a Tilted Airfoil.30which remains asymptotically 1 at large distances. By Cauchy's Theorem 7.48 coupled with formula (7.123), if C is a curve going once around the disk in a counter-clockwise direction, then i 1 dz = - 2
UCF - MATH - 5587
is analytic in the disk | z | 2 since its only singularity, at z = 3, lies outside the contour C. Therefore, by Cauchy's formula (7.135), we immediately obtain the integral ez dz = z2 - 2 z - 3 f (z) i dz = 2 i f (-1) = - . z+1 2eCCNote: Path independe
UCF - MATH - 5587
Chapter 12 Dynamics of Planar MediaIn previous chapters we studied the equilibrium configurations of planar media - plates and membranes - governed by the two-dimensional Laplace and Poisson equations. In this chapter, we analyze their dynamics, modeled
UCF - MATH - 5587
In this manner, we arrive at the basic conservation law relating the heat energy density and the heat flux vector w. As in our one-dimensional model, cf. (4.3), the heat energy density (t, x, y) is proportional to the temperature, so (t, x, y) = (x, y) u(
UCF - MATH - 5587
for the diffusion equation. See [35; p. 369] for a precise statement and proof of the general theorem. Qualitative Properties Before tackling examples in which we are able to construct explicit formulae for the eigenfunctions and eigenvalues, let us see w
UCF - MATH - 5587
Theorem 12.1. Suppose u(t, x, y) is a solution to the forced heat equation ut = u + F (t, x, y), for (x, y) , 0 &lt; t &lt; c,where is a bounded domain, and &gt; 0. Suppose F (t, x, y) 0 for all (x, y) and 0 t c. Then the global maximum of u on the set cfw_ (t, x
UCF - MATH - 5587
so there are no non-separable eigenfunctions . As a consequence, the general solution to the initial-boundary value problem can be expressed as a linear combination u(t, x, y) =m,n = 1cm,n um,n (t, x, y) =m,n = 1cm,n e- m,n t vm,n (x, y)(12.41)of
UCF - MATH - 5587
Let us start with the equation for q(). The second boundary condition in (12.50) requires that q() be 2 periodic. Therefore, the required solutions are the elementary trigonometric functions q() = cos m or sin m , where = m2 , (12.53)with m = 0, 1, 2, .