The solution in such cases is to use an alternative Clenshaw recurrence that incorporates ck ’s in an upward direction. The relevant equations are y−2 = y−1 = 0 1 [yk−2 − α(k, x)yk−1 − ck ], yk = β(k + 1, x) (k = 0, 1, . . . , N − 1) f(x) = cN FN (x) − β(N, x)FN−1 (x)yN−1 − FN (x)yN−2 () | Quadratic and Cubic Equations 183 The solution in such cases is to use an alternative Clenshaw recurrence that incorporates ck s in an upward direction. The relevant equations are y-2 y-1 0 yk p k 1 x vk-2 - a k x yk-i - ck k 0 1 . N -1 f x CNFn x - 3 N x Fn-i x yN-1 - Fn x yN-2 The rare case where equations - should be used instead of equations and can be detected automatically by testing whether the operands in the first sum in are opposite in sign and nearly equal in magnitude. 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 upward or downward direction. CITED REFERENCES AND FURTHER READING Abramowitz M. and Stegun . 1964 Handbook of Mathematical Functions Applied Mathematics Series Volume 55 Washington National Bureau of Standards reprinted 1968 by Dover Publications New York pp. xiii 697. 1 Gautschi W 1967 SIAM Review vol. 9 pp. 24-82. 2 Lakshmikantham V. and Trigiante D. 1988 TheoryofDifference Equations NumericalMethods and Applications San Diego Academic Press . 3 Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America pp. 20ff. 4 Clenshaw 1962 Mathematical Tables vol. 5 National Physical Laboratory London . Stationery Office . 5 Dahlquist G. and Bjorck A. 1974 Numerical Methods Englewood Cliffs NJ Prentice-Hall p. 111. Goodwin . ed. 1961 Modern Computing Methods 2nd ed. New York Philosophical Library p. 76. Quadratic and Cubic Equations The roots of simple algebraic equations can be viewed as being functions of the equations coefficients. We are taught these functions in elementary algebra. Yet surprisingly many people don t know the right way to solve a quadratic equation with two real roots or to obtain the roots of a cubic equation. There are two ways to write the solution of the quadratic equation Sample page from .
