Homework 1 Introduction to Computational Finance Spring 2019

Transcription

1 Homework 1 Introduction to Computational Finance Spring 2019 Solutions due Monday, 2/4/19 Answers to the homework problems and programming tasks should be sent via FSU s dropbox to kgallivan@fsu.edu before 11:59 PM on the due date. For problems with written solutions you may scan handwritten solutions and deposit the resulting pdf file in dropbox. Similarly, the report on the programming tasks should be sent as a pdf file. Do not sent Word files or any other text processing tool s input file. Code files should be included as described in Notes on Reporting on Programming Assignments. As with all homework assignments you are allowed and encouraged to consult the relevant literature. You are also expected to cite all literature that is used to generate your solutions and your solutions must make clear your understanding of the work cited. Programming Assignment Note this portion will not be graded and an example code is given. You are still expected to turn in examples of the use of the code to demonstrate conditioning, stability, and the effect of finite precision. 1.1.a During the semester you will develop a subroutine library of useful routines. These routines will be used throughout the semester. The first routine is used to check the floating point characteristics of your machine. Write and save a re-useable class that #DEFINE s a new floating point number type, e.g., FMFloat, that can be changed at compile time to either float or double and has the following methods: double epsilon (double) float epsilon (float) double huge (double) float huge (float) double tiny (double) float tiny (float) 1

2 that return the machine epsilon,the largest floating point number and the smallest floating point number. You can get the numbers that are required to be returned from the header file <float.h>. This class will make it possible for you to change the floating point data type used in your code for all appropriate variables at compile-time, i.e., without having to modify all of the variables and computations explicitly. Compare the information returned to what you know about IEEE floating point in single and double precision. Compare also the spacing, absolute and relative, on your machine to those of IEEE floating point in single and double precision. 1.1.b Using your class from the previous part of the problem, design and implement codes that: (i) verify that your double precision / single precision selections work; (ii) illustrate examples of stability (good and bad), conditioning from the notes or your readings in the literature. When discussing your examples demonstrating well-conditioned and ill-condition fuctions make sure to specifically address how you are dealing with the fact that you need the exact answer for both the original data and the perturbed data. Note that comparing single and double precision can provide useful insights but you it is also useful to introduce perturbations that are larger than the unit roundoffs of these precisions to assess conditioning and the effect of floating point errors when fewer digits are used. In particular, a crude but sometimes useful method of simulating t-digit decimal arithmetic is to use IEEE double precision as the true result and to modifiy the results of every computation with a random relative error. For example, A = B + C in an algorithm would be computed as D = B + C in double precision and the answer would be A (1 + d) where d is a random number between 1 and 1 times 10 t. Similarly when demonstrating the difference between the behavior of: a well-conditioned problem solved by an unstable algorithm; a well-conditioned problem solved by an stable algorithm; a ill-conditioned problem solved by an stable algorithm; discuss the choices made for precision used in your empirical demonstrations. You must also discuss what makes the algorithms and problems unstable/stable and well/ill conditioned respectively. Solution: A C++ code that accomplishes the tasks given uses the following class and include files: 2

3 /* * FloatingPoint.h * floatingpoint * #ifndef FLOATING_POINT_REQ #define _FLOATING_POINT_REQ #include <float.h> class FPEnvironment { public: static double epsilon(double) {return DBL_EPSILON;}; static float epsilon(float) {return FLT_EPSILON;}; static double maxval(double) {return DBL_MAX;}; static float maxval(float) {return FLT_MAX;}; static double minval(double) {return DBL_MIN;}; static float minval(float) {return FLT_MIN;}; }; #endif /* * Precision.h * */ /* /* Comment out the appropriate line below to specify the precision of an rtfloat */ // typedef float rtfloat; typedef double rtfloat; A simple code that uses the types to do nothing important is: #include <iostream> #include "Precision.h" #include "FloatingPoint.h" #include <math.h> 3

4 int main (int argc, char * const argv[]) { rtfloat a = 0.0; rtfloat b = 3.0; rtfloat atol = 1.0e-37; rtfloat rtol = 1.0e-9; // // // Check input parameters // // if( rtol < FPEnvironment::epsilon(rTol) ) { std::cout << "Relative tolerance set too low, resetting to 2*epsilon" << std::endl; rtol = 2*FPEnvironment::epsilon(rTol); } } This portion will be graded. Problem 1.2 Written Exercises 1.2.a. Suppose x R and y R with x < y. Is it always true that fl(x) < fl(y) in any standard model floating point system? 1.2.b. Suppose x, y and z are floating point numbers in a standard model floating point arithmetic system. Is floating point arithmetic associative, i.e., is it true that (x op (y op z)) = ((x op y) op z)? 1.2.c. Is floating point arithmetic distributive, i.e., is it true that fl(fl(x + z) y) = fl(fl(fl(x y) + fl(y z)))? 1.2.d. Suppose x and y are two floating point numbers in a system F(β, t, L, U) with opposite signs. How close do x and y have to be in magnitude in order for the result of the floating point computation to be exact? (x + y) 4

5 Problem 1.3 This problem considers the roots of the quadratic equation with a single parameter β > 1 x 2 + 2βx + 1. Define the vector-valued function that maps β to the two roots x + (β) and x (β) f : R R 2, β 1.3.a What happens to the roots as β? ( ) x+ (β) = x (β) ( ) β + β2 1 β β b For β > 1 derive an approximation of the relative condition number for the vector of roots f(β) when β is perturbed slightly. Note that since f is a vectorvalued function a vector norm must be used. For your analysis use the standard Euclidean 2-norm, i.e., v = ( ν1 ν 2 ) v 2 = ν ν 2 2. The absolute value be used as the norm of the scalars β and β in R. That is we have f(β + β) f(β) 2 f(β) 2 κ rel β β. 1.3.c Use the condition number to explain the conditioning of the vector of roots for β > 1, i.e., is it well-conditioned anywhere on the interval, is it ill-conditioned anywhere on the interval? 1.3.d Can your approach to approximating the relative condition number of the vector of roots be used for β = 1? Justify your answer. Problem 1.4 Consider finding the roots x 1 and x 2 of a quadratic equation ax 2 + bx + c via the quadratic formula b ± b 2 4ac 2a (1.4.a) Identify the computations and situations with the values of a, b, c where cancellation might take place. (There are two.) 5

6 (1.4.b) Consider the quadratic and its roots x x = 0 x 1 = x 2 = Use the quadratic formula with floating point arithmetic defined by β = 10 and t = 4 with rounding (you may assume that L and U give enough range to handle any exponents needed for this problem) to solve for x 1 and x 2. (1.4.c) Did cancellation affect the results? Explain. Problem 1.5 Consider the following numbers: a. Express the numbers as floating point numbers with β = 10 and t = 4 using rounding to even and using chopping. 1.5.b. Express the numbers as floating point numbers with in single precision IEEE format using rounding to even. It is strongly recommended that you implement a program to do this rather than computing the representation manually. 1.5.c. Calculate the relative error for each number and verify it satisfies the bounds implied by the floating point system used. 6