Roots of equations, minimization, numerical integration

Transcription

1 Roots of equations, minimization, numerical integration Alexander Khanov PHYS6260: Experimental Methods is HEP Oklahoma State University November 1, 2017

2 Roots of equations Find the roots solve equation f (x) = 0 find x, a < x < b, such that f (x) = 0 Multidimensional case: solve f (x) = 0 inside some region of x values In general not a trivial problem: usually not possible to be done analytically except for simple cases experimentalists hate to solve anything more complicated than a quadratic equation impractical if one needs to probe many different functions (e.g. polynomials of different degrees) even less practical if the function value is a result of some numerical calculation (no explicit formula) A. Khanov (PHYS6260, OSU) PHYS /1/17 2 / 13

3 Roots of equations (2) Before looking for roots, one needs to make sure they exist if f (x) is continuous and f (a) > 0, f (b) < 0, then it has a root between a and b This is not necessarily trivial multidimensional case: continuity in each argument does not imply multivariate continuity numerically computed functions: difficult to prove anything It s OK to find the root(s) approximately: instead of finding x such that f (x) = 0, find x such that x x < ε for a reasonably small ε whatever comes out of a computer usually has an uncertainty due to finite number of digits in the floating point number representation it s important not to set ε smaller than computing precision (true for any numerical method) A. Khanov (PHYS6260, OSU) PHYS /1/17 3 / 13

4 Simple methods to find the root Bisection: divide the current interval in half, pick the half where f (x) has opposite signs on its ends simple, the error on x decreases by a factor of 2 on each step, not very fast False position (variation of secant method): draw a straight line through (a, f (a)) and (b, f (b)), find the intersection with the y = 0 axis, proceed as in bisection Converges faster than bisection, but the interval width may or may not approach zero (if it doesn t, it s not clear when to stop) Both methods can be easily extended to multidimensional case A. Khanov (PHYS6260, OSU) PHYS /1/17 4 / 13

5 Roots and minimization The two problems can be solved in the same way: finding a root of f (x) is equivalent of minimizing f (x), or (f (x)) 2 if analytic behavior is desirable x is not analytic at 0 (no derivative, no Taylor series) Example: f (x) = x has a root x = 0 and (f (x)) 2 = x 2 has a minimum at x = 0 (equal to 0) The advantages of replacing root finding with minimum finding become significant in the multidimensional case A. Khanov (PHYS6260, OSU) PHYS /1/17 5 / 13

6 Newton s method Start from zero approximation x 0, proceed according to the formula x k+1 = x k f (x k) f (x k ) Advantages: converges really fast usually the number of correct digits doubles at every step this is called quadratic convergence (not true for roots with multiplicity > 1) cf. bisection method, a new correct (binary) digit is added at every step linear convergence Disadvantages: need to know the first derivative (may prove nontrivial for computed function); sometimes it does not converge at all f (x) = x 3 2x + 2, f (x) = 3x 2 2 x 0 = 0, f (0) = 2, f (0) = 2, x 1 = = 1 x 1 = 1, f (1) = 1, f (1) = 1, x 2 = = 0 A. Khanov (PHYS6260, OSU) PHYS /1/17 6 / 13

7 Newton s algorithm: multidimensional case Newton s method can be generalized to miltidimensional case of n functions f = f 1,..., f n of n variables x = x 1,..., x n Start from zero approximation x 0, proceed according to the formula x k+1 = x k J(x k ) 1 f(x k ) ( ) fi where J = is the jacobian x j A modification of the method (Newton-Gauss algorithm) can be used in case when the number of functions n is larger than the number of variables m, in this case the procedure x k+1 = x k (J(x k ) T J(x k )) 1 J(x k ) T f(x k ) finds the minimum of n (f i (x)) 2 non-linear least squares problem i=1 A. Khanov (PHYS6260, OSU) PHYS /1/17 7 / 13

8 Problems of root finding All algorithms are in trouble when there are many roots No really good advice except trying several points and seeing where it converges f(x)=x -x A. Khanov (PHYS6260, OSU) PHYS /1/17 8 / 13

9 Minimization Minimization of f (x) = maximization of f (x) (the same problem) Since at minimum f (x) = 0, minimization = finding the root of f (x) = 0 Gradient descent (multidimensional case): make steps proportional to the megative of the gradient A lot of other methods, more or less heuristic A. Khanov (PHYS6260, OSU) PHYS /1/17 9 / 13

10 MINUIT The only program you need to do the minimization The rules of the game are the following: 1 The function F (x) is assumed not to be known analytically, but is specified by giving its value at any point x. 2 The allowed values of the variable x may be restricted to a certain range, in which case one speaks of constrained minimization. Here only unconstrained problems are refered to. 3 In some cases additional information about the function F may be available, such as the numerical values of the derivative F / x at any point x. Such knowledge cannot in general be assumed, but should be used when possible. 4 The function F (x) is repeatedly evaluated at different points x until its minimum value is attained. (from the MINUIT tutorial) A. Khanov (PHYS6260, OSU) PHYS /1/17 10 / 13

11 1d numerical integration Do it analytically best, but not always possible Interpolate + integrate the interpolation Linear interpolation trapezoidal rule good: ( b f (a) + f (b) f (x) dx = x 2 better: b a a f (x) dx = x (x 1 = a, x n = b) ( f (a) + f (b) 2 n 1 + ) f (x i ) + O( x 2 ) n 1 + ) f (x i ) i=2 i=2 + x 2 12 (f (a) f (b))+o( x 3 ) Any interpolation works, but beware of interpolation problems (Runge s effect etc.) A. Khanov (PHYS6260, OSU) PHYS /1/17 11 / 13

12 Improper integrals Integrals with infinite limits cut the integration at some point where the function is small small can be usually estimated from the fact that f (x) dx < φ(x) dx if f (x) < φ(x) for all x > x 0 x 0 x 0 change variables to transform to finite f (a) but b a f (x) dx exists and is finite integrate by parts, change variables to transform to finite if not possible, do the numerical integration but use uneven spacing dx = 2 x = ξ, x = ξ 2, dx = 2ξ dξ 2dξ 0 x 0 A. Khanov (PHYS6260, OSU) PHYS /1/17 12 / 13

13 Multidimensional integrals Usually the function is not difficult to calculate, but the integration limits are tricky Typical approach: Monte Carlo techinique A. Khanov (PHYS6260, OSU) PHYS /1/17 13 / 13