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5.6 Quadratic and Cubic Equations183The solution in such cases is to use an alternative Clenshaw recurrence thatincorporates ck ’s in an upward direction. The relevant equations aref(x) = cN FN (x) − β(N, x)FN−1 (x)yN−1 − FN (x)yN−2(5.5.25)(5.5.26)(5.5.27)The rare case where equations (5.5.25)–(5.5.27) should be used instead ofequations (5.5.21) and (5.5.23) can be detected automatically by testing whetherthe operands in the first sum in (5.5.23) are opposite in sign and nearly equal inmagnitude.
Other than in this special case, Clenshaw’s recurrence is always stable,independent of whether the recurrence for the functions Fk is stable in the upwardor downward direction.CITED REFERENCES AND FURTHER READING:Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 byDover Publications, New York), pp. xiii, 697. [1]Gautschi, W. 1967, SIAM Review, vol.
9, pp. 24–82. [2]Lakshmikantham, V., and Trigiante, D. 1988, Theory of Difference Equations: Numerical Methodsand Applications (San Diego: Academic Press). [3]Acton, F.S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington: Mathematical Association of America), pp. 20ff.
[4]Clenshaw, C.W. 1962, Mathematical Tables, vol. 5, National Physical Laboratory (London: H.M.Stationery Office). [5]Dahlquist, G., and Bjorck, A. 1974, Numerical Methods (Englewood Cliffs, NJ: Prentice-Hall),§4.4.3, p. 111.Goodwin, E.T. (ed.) 1961, Modern Computing Methods, 2nd ed. (New York: Philosophical Library), p.
76.5.6 Quadratic and Cubic EquationsThe roots of simple algebraic equations can be viewed as being functions of theequations’ coefficients. We are taught these functions in elementary algebra. Yet,surprisingly many people don’t know the right way to solve a quadratic equationwith two real roots, or to obtain the roots of a cubic equation.There are two ways to write the solution of the quadratic equationax2 + bx + c = 0(5.6.1)with real coefficients a, b, c, namelyx=−b ±√b2 − 4ac2a(5.6.2)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).y−2 = y−1 = 01[yk−2 − α(k, x)yk−1 − ck ],yk =β(k + 1, x)(k = 0, 1, . . . , N − 1)184Chapter 5.andx=Evaluation of Functions2c√−b ± b2 − 4ac(5.6.3)q≡−ip1hb + sgn(b) b2 − 4ac2(5.6.4)Then the two roots arex1 =qax2 =andcq(5.6.5)If the coefficients a, b, c, are complex rather than real, then the above formulasstill hold, except that in equation (5.6.4) the sign of the square root should bechosen so as to makep(5.6.6)Re(b* b2 − 4ac) ≥ 0where Re denotes the real part and asterisk denotes complex conjugation.Apropos of quadratic equations, this seems a convenient place to recall thatthe inverse hyperbolic functions sinh−1 and cosh−1 are in fact just logarithms ofsolutions to such equations,px2 + 1pcosh−1 (x) = ± ln x + x2 − 1sinh−1 (x) =ln x +(5.6.7)(5.6.8)Equation (5.6.7) is numerically robust for x ≥ 0.
For negative x, use the symmetrysinh−1 (−x) = − sinh−1 (x). Equation (5.6.8) is of course valid only for x ≥ 1.For the cubic equationx3 + ax2 + bx + c = 0(5.6.9)with real or complex coefficients a, b, c, first computeQ≡a2 − 3b9andR≡2a3 − 9ab + 27c54(5.6.10)If Q and R are real (always true when a, b, c are real) and R2 < Q3 , then the cubicequation has three real roots. Find them by computingθ = arccos(R/pQ3 )(5.6.11)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).If you use either (5.6.2) or (5.6.3) to get the two roots, you are asking for trouble: Ifeither a or c (or both) are small, then one of the roots will involve the subtractionof b from a very nearly equal quantity (the discriminant); you will get that root veryinaccurately.
The correct way to compute the roots is1855.6 Quadratic and Cubic Equationsa3a3(5.6.12)(This equation first appears in Chapter VI of François Viète’s treatise “De emendatione,” published in 1615!)Otherwise, computehi1/3pA = − R + R 2 − Q3(5.6.13)where the sign of the square root is chosen to makepRe(R* R2 − Q3 ) ≥ 0(5.6.14)(asterisk again denoting complex conjugation). If Q and R are both real, equations(5.6.13)–(5.6.14) are equivalent toi1/3hp(5.6.15)A = −sgn(R) |R| + R2 − Q3where the positive square root is assumed.
Next computeQ/A(A 6= 0)B=0(A = 0)(5.6.16)in terms of which the three roots arex1 = (A + B) −a3(5.6.17)(the single real root when a, b, c are real) and√a13(A − B)x2 = − (A + B) − + i232(5.6.18)√a13(A − B)x3 = − (A + B) − − i232(in that same case, a complex conjugate pair). Equations (5.6.13)–(5.6.16) arearranged both to minimize roundoff error, and also (as pointed out by A.J. Glassman)to ensure that no choice of branch for the complex cube root can result in thespurious loss of a distinct root.If you need to solve many cubic equations with only slightly different coefficients, it is more efficient to use Newton’s method (§9.4).CITED REFERENCES AND FURTHER READING:Weast, R.C.
(ed.) 1967, Handbook of Tables for Mathematics, 3rd ed. (Cleveland: The ChemicalRubber Co.), pp. 130–133.Pachner, J. 1983, Handbook of Numerical Analysis Applications (New York: McGraw-Hill), §6.1.McKelvey, J.P. 1984, American Journal of Physics, vol. 52, pp. 269–270; see also vol. 53,p. 775, and vol. 55, pp. 374–375.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).in terms of which the three roots are paθ−x1 = −2 Q cos33pθ + 2π−x2 = −2 Q cos3pθ − 2π−x3 = −2 Q cos3186Chapter 5.Evaluation of Functions5.7 Numerical Derivativesf 0 (x) ≈f(x + h) − f(x)h(5.7.1)practically suggests the program: Pick a small value h; evaluate f(x + h); youprobably have f(x) already evaluated, but if not, do it too; finally apply equation(5.7.1).
What more needs to be said?Quite a lot, actually. Applied uncritically, the above procedure is almostguaranteed to produce inaccurate results. Applied properly, it can be the right wayto compute a derivative only when the function f is fiercely expensive to compute,when you already have invested in computing f(x), and when, therefore, you wantto get the derivative in no more than a single additional function evaluation.
In sucha situation, the remaining issue is to choose h properly, an issue we now discuss:There are two sources of error in equation (5.7.1), truncation error and roundofferror. The truncation error comes from higher terms in the Taylor series expansion,11f(x + h) = f(x) + hf 0 (x) + h2 f 00 (x) + h3 f 000 (x) + · · ·26(5.7.2)whence1f(x + h) − f(x)= f 0 + hf 00 + · · ·h2(5.7.3)The roundoff error has various contributions. First there is roundoff error in h:Suppose, by way of an example, that you are at a point x = 10.3 and you blindlychoose h = 0.0001. Neither x = 10.3 nor x + h = 10.30001 is a number withan exact representation in binary; each is therefore represented with some fractionalerror characteristic of the machine’s floating-point format, m , whose value in singleprecision may be ∼ 10−7 .
The error in the effective value of h, namely the differencebetween x + h and x as represented in the machine, is therefore on the order of m x,which implies a fractional error in h of order ∼ m x/h ∼ 10−2 ! By equation (5.7.1)this immediately implies at least the same large fractional error in the derivative.We arrive at Lesson 1: Always choose h so that x + h and x differ by an exactlyrepresentable number.
This can usually be accomplished by the program stepstemp = x + hh = temp − x(5.7.4)Some optimizing compilers, and some computers whose floating-point chips havehigher internal accuracy than is stored externally, can foil this trick; if so, it isusually enough to declare temp as volatile, or else to call a dummy functiondonothing(temp) between the two equations (5.7.4). This forces temp into andout of addressable memory.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Imagine that you have a procedure which computes a function f(x), and nowyou want to compute its derivative f 0 (x). Easy, right? The definition of thederivative, the limit as h → 0 of.
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