c 2002 Society for Industrial and Applied Mathematics 

SIAM REVIEW Vol. 44, No. 2, pp. 227–233

Explicit Solutions for Transcendental Equations∗ Rogelio Luck† James W. Stevens† Abstract. A simple method to formulate an explicit expression for the roots of any analytic transcendental function is presented. The method is based on Cauchy’s integral theorem and uses only basic concepts of complex integration. One convenient method for numerically evaluating the exact expression is presented. The application of both the formulation and evaluation of the exact expression is illustrated for several classical root finding problems. Key words. analytic functions, transcendental equations, Cauchy integral theorem AMS subject classifications. 30E20, 65H05 PII. S0036144501386111 Notation. λ wavelength B Biot number C2 second radiation constant, 1.439 × 104 µm*K f functions of z h, R center and radius of a circle in the complex plane S Stefan number w complex function z independent variable value of z at a singularity zo

1. Introduction. The determination of roots of transcendental functions is a problem commonly encountered in a broad spectrum of engineering applications (e.g., Fettis (1976), Siewert and Burniston (1972), Siewert and Phelps (1978)). There exist a wide variety of numerical methods that are useful for approximating the solution to any desired degree of accuracy. From a practical standpoint, such root finding algorithms are generally straightforward to use and provide an adequate approach for determining values for the roots. Nevertheless, it is occasionally useful to have an exact mathematical solution to the problem under consideration. For example, an explicit expression for the root allows development of analytical derivatives for uncertainty analyses and sensitivity studies. In many cases analytical derivatives provide much greater insight into the problem than numerical derivatives. Explicit expressions are also useful for checking convergence of approximate root finding techniques. HajiSheikh and Beck (2000) described applications of the closed form expressions that they present from several analytical heat transfer problems. This paper describes a simple method of formulating exact explicit solutions for the roots of analytic tran∗ Received by the editors March 7, 2001; accepted for publication (in revised form) December 17, 2001; published electronically May 1, 2002. http://www.siam.org/journals/sirev/44-2/38611.html † Department of Mechanical Engineering, Mississippi State University, P.O. Drawer ME, Mississippi State, MS 39762 ([email protected], [email protected]).

227

228

ROGELIO LUCK AND JAMES W. STEVENS

scendental functions using Cauchy’s integral theorem from complex analysis. The method is developed and several examples are described. General descriptions of common root finding techniques are available in most textbooks in the area of numerical analysis, e.g., Burden and Faires (1985). Leathers and McCormick (1996) addressed a different methodology for obtaining explicit solutions to several transcendental problems arising in heat transfer. The general approach used was based on methods of Muskhelishvili (1953) and was developed by Burniston and Siewert (1973). This approach “depends on formulating an appropriate Riemann problem of complex variable theory and then expressing the solution of the transcendental equation in terms of a canonical solution of that problem” (Leathers and McCormick, 1996). In some cases this approach is difficult to implement and results in expressions that are cumbersome to evaluate. This paper describes a more elemental method that uses only the most basic concepts of complex integration and that can be easily applied to a large variety of problems. 2. Development. The approach presented here determines the roots of a function by locating the singularities of the reciprocal of the function. Cauchy’s theorem (Strang, 1986) states that if a function is analytic in a simple connected region containing the closed curve C, the path integral of the function around C is zero. On the other hand, if a function, f (z), contains a single singularity at zo somewhere inside C but is analytic elsewhere in the region, then the singularity can be removed by multiplying f (z) by (z − zo ), i.e., by a pole-zero cancellation. Cauchy’s theorem implies that the path integral of the new function around C must be zero:  (1) (z − zo )f (z) dz = 0. C

Evaluating this integral yields a first-order polynomial in zo with constant coefficients. Solving the polynomial for zo yields the location of the singularity:  zf (z) dz zo = C (2) . f (z) dz C

Equation (2) is an explicit expression for the singularity of the function f (z). A root finding problem may be recast as a singularity at the root, and (2) yields the desired root. As a practical matter for determining roots, the evaluation of (2) may or may not compete well with approximate numerical techniques for root finding. However, the method and ease of evaluation is an entirely separate issue from the derivation of an exact, explicit solution. The notable result is that (2) provides just such an explicit solution for the root. This expression, (2), can be evaluated over any closed path and with any technique, analytical or numerical, that is convenient. One strategy for evaluation of (2) that results in a particularly straightforward calculation uses a circle in the complex plane that circumscribes the root. The closed curve C may then be described as a circle in the complex plane with center h and radius R: (3)

z = h + Reiθ ,

dz = iReiθ dθ.

The values of h and R do not matter as long as the circle circumscribes the root. Cauchy’s principle of the argument may be used to determine the number of roots

EXPLICIT SOLUTIONS FOR TRANSCENDENTAL EQUATIONS

229

enclosed by the path C. Equation (2) becomes 

(4)

 i2θ w(θ)e dθ  0  , zo = h + R    2π  iθ w(θ)e dθ 2π

0

where w(θ) = f (h + Reiθ ). The structure of (4) makes evaluation very accessible. The definition of the nth coefficient in a complex exponential Fourier series is (5)

1 An = 2π





x(t)eint dt, 0

where x(t) is the function to be represented by the series. It can be seen that the term in brackets in (4) is equal to the ratio of the second Fourier series coefficient to the first for the function w(θ). Fourier series coefficients may be calculated easily using standard complex fast Fourier transform (FFT) functions found in most mathematical software packages. Notice that, provided that f (z) is analytic at h, multiplying f (z) by a factor of (z − h) = Reiθ will not change the location of the singularities of f (z). This implies that for a given singularity the term in brackets is also equal to any ratio of the (j + 1) to the jth Fourier series coefficients of w(θ), with j ≥ 1. 3. Examples. For a simple illustration, (4) will be used to derive an explicit expression for the roots of the transcendental equation (6)

z tan(z) = B.

This equation arises from the infinite series solution to the one-dimensional transient heat conduction problem with one adiabatic and one convective boundary condition. A similar equation comes from the solution of a longitudinal vibration problem in a uniform bar with one fixed and one attached mass boundary condition. Positive roots of the equation are required. From a consideration of the functions tan(z) and 1/z (see Figure 1), it is apparent that for the nth root, a suitable choice for the closed path C is a circle of radius R = π/4 centered at hn = (n − 3/4)π. A straightforward way to configure the equation to provide a singularity is to simply use the inverse of a rearranged equation (6): (7)

f (z) =

1 . z sin(z) − B cos(z)

Any other convenient configuration (e.g., 1/[1 − e(x tan(x)−B) ]) could be used to create the singularity required for (1). Applying (4) results in an explicit expression for the nth root. This expression can be evaluated simply by calculating the FFT as described earlier, or via numerical integration. The first root for B = 2, which is 1.07687 . . . , can be evaluated correct to 11 decimal places with an FFT using 32 points. Figure 2 illustrates the effect of the number of points used in the FFT on the accuracy of the result. Each point requires one function evaluation. The number of function evaluations required for a secant method root finding algorithm using (6) is included simply as a point of interest. Clearly the secant method algorithm surpasses the FFT in computational efficiency in generating an approximation to the exact

230

ROGELIO LUCK AND JAMES W. STEVENS

2 1.8 1.6 1.4 1.2 1

2/z

tan(z)

0.8 0.6

tan(z)

1/z

0.4 1st root, Bi=2 1.07687...

0.2 0

π/2

π

z

3π/2

Fig. 1 Roots of z tan(z) = B.

10 1 0.1 0.01 1 .10

3

1 .10

4

1 .10

5

1 .10

6

percent

1 .10

7

error in

1 .10

8

root

1 .10

9

1 .10

10

1 .10

11

1 .10

12

1 .10

13

1 .10

14

1 .10

15

1 .10

16

1 .10

17

secant method

0

4

8

12

16

20

24

28

number of function evaluations Fig. 2 Convergence of FFT method of evaluating exact formulation of roots.

32

231

EXPLICIT SOLUTIONS FOR TRANSCENDENTAL EQUATIONS

answer; however, (2) and (4) provide the exact answer regardless of the method used to evaluate it. Function derivatives are commonly used in uncertainty calculations and other analytical applications. The calculation of an explicit exact derivative, dzo /dB, is straightforward from (4) and (7):

F1 (w)F2 (w ) − F2 (w)F1 (w ) dzo =R (8) , dB F1 (w)2 where 



xeinθ dθ,

Fn (x) = 0

(9)

w = f (z), w =

dw − cos(z) . = dB (z sin(z) − B cos(z))2

For the first root with B = 2, the value of the derivative is dzo /dB = 0.1504 . . . . Figure 3 illustrates the error associated with three different evaluations of the derivative using an FFT. Using 32 points in the FFT results in an error of less than 10−11 . For comparison, Figure 3 includes the error in the approximate derivative calculated numerically from a secant method root finding algorithm. The numerical derivative was approximated by (10)

SR(B + ε) − SR(B − ε) dzo ≈ , dB 2ε

where SR(B) represents the secant method root finding algorithm. The error is a function of the perturbation size, ε. Below ε ≈ 10−5 the round-off error in the numerical approximation impeded further improvement in the approximation. A second illustrative example comes from Wien’s law. The wavelength of maximum emissive power for black-body radiation can be obtained from (5 − z)ez = 5, where z = C2 /λT . While this equation needs to be solved only once, the solution provides a nice illustration of the technique of this paper. The function 5/(5 − z) intersects ez at two locations. One is the trivial solution z = 0, and the other intersection must take place to the left of z = 5, since 5/(5 − z) approaches infinity as z approaches 5. Selecting f (z) = 1/((5 − z)ez − 5), the required pole-zero cancellations are achieved by multiplying by (z − 0)(z − zo ). As a result, the integrand of (1) becomes z(z − zo )f (z). Using R = 5 and h = 0, integrating and solving for zo leads to modifying (4) to be 

(11)

 i3θ w(θ)e dθ  0  , zo = R    2π  i2θ w(θ)e dθ 2π

0

where w(θ) = f (h + Reiθ ) =

1 [(5 − (h +

Reiθ ))e(h+Reiθ )

− 5]

.

232

ROGELIO LUCK AND JAMES W. STEVENS 0.01

.

3

.

4

.

5

.

6

.

7

.

8

.

9

8 evaluations 8 points

1 10

1 10

16 evaluations 16 points

1 10

1 10

absolute value of error

1 10

secant method

1 10

1 10

.

10

.

11

1 10

32 evaluations

1 10

.

1 10

32 points

12

.

1 10

9

.

1 10

8

.

1 10

7

.

1 10

6

.

1 10

5

.

1 10

4

.

1 10

3

0.01

0.1

1

epsilon Fig. 3

Accuracy of the derivative evaluated numerically with the secant method and with the FFT method.

The term in brackets is equal to the ratio of the third to the second Fourier series coefficients of w(θ). As before, this term is also equal to the ratio of the (j + 1) to the jth Fourier series coefficients of w(θ), with j ≥ 2. Evaluation of this expression with R = 5 yields zo = 4.9651 . . . . A third illustration treats the equation (12)

√ 2 zez erf(z) = S/ π.

For solidification of a pure liquid, initially at its phase change temperature in a semi-infinite region, the temperature profile in the liquid, the temperature profile in the solid, and the location of the interface need to be determined as functions of time. Roots of the above transcendental equation must be determined for the solution of this problem. Since the left side of the equation is a parabolic-shaped, even function, there will be two solutions of opposite sign but equal magnitude. This makes it possible to focus on solutions on the positive real axis without losing generality. On the real axis, erf(z) ≈ 1 for z ≥ 2, and the left side of the equation is approximately 100 when√z = 2. Therefore, one can choose a circle with h = 1 and R = 1 for S ≤ 100∗ π. For larger S, a rough approximation to the solution can be obtained √ by letting erf(z) ≈ 1, neglecting ln(z) when solving for z to obtain z  sqrt(ln(S/ π)). √ Thus, h can be set to h = sqrt(ln(S/ π)) and R is set to R = 0.2 to account for errors in the approximation. As before, f (z) is just the reciprocal of the equation

EXPLICIT SOLUTIONS FOR TRANSCENDENTAL EQUATIONS

233

√ 2 under consideration, f (z) = 1/(zez erf(z) π − S), making w(θ) =

1 (h +

Reiθ )e(h+Reiθ )2 erf(h

, √ + Reiθ ) π − S

and the desired solution is given by (4). Conclusion. A simple procedure for determining explicit solutions to a number of transcendental problems has been presented. Numerical evaluation of solutions is readily accessible, requiring only a complex Fourier transform. While the computational efficiency of this procedure would not be expected to rival that of traditional approximate root finding techniques, the procedure is conceptually simple, provides an exact explicit expression for the roots, and can easily be implemented with the tools commonly available in commercial mathematical packages. REFERENCES R. L. Burden and J. D. Faires (1985), Numerical Analysis, PWS, Boston. E. E. Burniston and C. E. Siewert (1973), The use of Riemann problems in solving a class of transcendental equations, Proc. Cambridge Philos. Soc., 73, pp. 111–118. H. E. Fettis (1976), Complex roots of sin(z) = az, cos(z) = az, and cosh(z) = az, Math. Comp., 30 (135), pp. 541–545. A. Haji-Sheikh and J. V. Beck (2000), An efficient method of computing eigenvalues in heat conduction, Numerical Heat Transfer B, 38, pp. 133–156. R. A. Leathers and N. J. McCormick (1996), Closed-form solutions for transcendental equations of heat transfer, ASME J. Heat Transfer, 118, pp. 970–973. N. I. Muskhelishvili (1953), Singular Integral Equations, Noordhoff, Groningen, The Netherlands. C. E. Siewert and E. E. Burniston (1972), An exact analytical solution of Kepler’s equation, Celestial Mechanics, 6, pp. 294–304. C. E. Siewert and J. S. Phelps (1978), On solutions of a transcendental equation basic to the theory of vibrating plates, J. Comput. Appl. Math., 4, pp. 37–39. G. Strang (1986), Introduction to Applied Mathematics, Wellesley-Cambridge, Wellesley, MA.

Explicit Solutions for Transcendental Equations - SIAM - Society for ...

Key words. analytic functions, transcendental equations, Cauchy integral ... simple method of formulating exact explicit solutions for the roots of analytic tran-.

452KB Sizes 4 Downloads 255 Views

Recommend Documents

On multiple solutions for multivalued elliptic equations ...
istence of multiple solutions for multivalued fourth order elliptic equations under Navier boundary conditions. Our main result extends similar ones known for the ...

Nonlinear Ordinary Differential Equations - Problems and Solutions ...
Page 3 of 594. Nonlinear Ordinary Differential Equations - Problems and Solutions.pdf. Nonlinear Ordinary Differential Equations - Problems and Solutions.pdf.

Siam Makro
Aug 9, 2017 - กําไรสุทธิขอ งMAKRO ใน 2Q60 อยู ที่1.23 พันล านบาท (+. 9% YoY, -24% QoQ) ต่ํากว าประมาณการของเรา 18% และ. ต่ํากà¸

dimensionality of invariant sets for nonautonomous processes - SIAM
Jan 21, 1992 - by various authors for a number of dissipative nonlinear partial differential equations which are either autonomous or are ... fractal dimensions of invariant sets corresponding to differential equations of the above type, subject to .

An optimal explicit time stepping scheme for cracks ...
of element degrees of freedom (in space and time as the crack is growing); ...... Réthoré J., Gravouil A., Combescure A. (2004) Computer Methods in Applied.

Improved explicit estimates on the number of solutions ...
May 12, 2004 - computer science, with many applications. ... [GL02b, Theorem 4.1]) the following estimate on the number N of q–rational points of an ... q is an absolutely irreducible Fq– definable plane curve of degree δ > 0 and the ...

Improved explicit estimates on the number of solutions ...
May 12, 2004 - solutions of equations over a finite field 1. A. Cafurea,b .... We shall denote by I(V ) ⊂ Fq[X1,...,Xn] its defining ideal and by Fq[V ] its coordinate ...

Siam Makro - Settrade
Aug 9, 2017 - กําไรสุทธิขอ งMAKRO ใน 2Q60 อยู ที่1.23 พันล านบาท (+. 9% YoY, -24% QoQ) ต่ํากว าประมาณการของเรา 18% และ. ต่ํากà¸

Newton's method for generalized equations
... 2008 / Accepted: 24 February 2009 / Published online: 10 November 2009 ..... Then there is a unique x ∈ Ba(¯x) satisfying x = Φ(x), that is, Φ has a unique.

Siam Global House - Settrade
May 7, 2018 - และ iii) GPM เพิ่มขึ้นจากสัดส่วนยอดขายสินค้า house brand. ที่เพิ่ม .... Figure 7: House brand accounts for 13% of total sale. Number of store, stores.

Society for Romanian Studies - WordPress.com
European Cultural Capital in spite of the brutal heat wave (95 degree and 95 ... We are equally grateful to the National Arts University in Bucharest ..... Metropolitan College or New York on Saturday,. December 1st .... Hall Studio 10), with a paper

siam commercial bank - efinanceThai
4 days ago - Management Plc. 11.56. % ... asset Management Plc. 11.56. %. Stock: .... management, custodial services, credit ..... Auto, Media, Health Care.

Society for Romanian Studies - WordPress.com
For the best part of a year I worked with Matt Ciscel—the committee chair—, .... generation and those who are most comfortable with new media and ... York, for his paper “'We Are the Losers of Socialism': Tuberculosis, Social ... 2016 is James

Explicit formulas for repeated games with absorbing ... - Springer Link
Dec 1, 2009 - mal stationary strategy (that is, he plays the same mixed action x at each period). This implies in particular that the lemma holds even if the players have no memory or do not observe past actions. Note that those properties are valid

047143115X.Wiley.Solaris Solutions for System Administrators.pdf ...
047143115X.Wiley.Solaris Solutions for System Administrators.pdf. 047143115X.Wiley.Solaris Solutions for System Administrators.pdf. Open. Extract. Open with.

An abstract factorization theorem for explicit substitutions
We show how to recover standardization by levels, we model both call-by-name and ... submitted to 23rd International Conference on Rewriting Techniques and ...

Explicit mean-field radius for nearly parallel vortex ...
Sep 12, 2007 - Astro. Fluid Dyn. 94, 177. Lim, C. C.: 2006, in Proc. IUTAM Symp., Plenary Talk in Proc. IUTAM Symp., Springer-Verlag,. Steklov Inst., Moscow. Lim, C. C. and Assad, S. M.: 2005, R & C Dynamics 10, 240. Lim, C. C. and Nebus, J.: 2006, V

Mathematics of the Transcendental
When a Notices editor asked me to review Badiou's book [2] I objected on the grounds that I am no philosopher, which only strengthened her determination. Here then is a mathematician's review of a philosopher's mathematics book. Alain Badiou (born 19