Physics 234: Lab 7 Tuesday, March 2, 2010 / Thursday, March 4, 2010

Transcription

1 Physics 234: Lab 7 Tuesday, March 2, 2010 / Thursday, March 4, Use the curl command to download from the class website everything you ll need for the lab. $ WEBPATH= $ curl $WEBPATH/Lab7.pdf -O $ curl $WEBPATH/src/Lab7.tar.gz -O $ evince Lab7.pdf& $ tar xzf Lab7.tar.gz $ cd Lab7 2. The program fp.cpp is designed to showcase the rules for floating point arithmetic in particular, how the special values inf, -inf, and nan are generated and how they propagate through arithmetic calculations. Make the necessary changes to the program so that it produces the output shown below: $ make fp g++ -o fp fp.cpp -O2 -ansi -pedantic -Wall -lm $./fp Floating point division: 1 / 0 = inf -1 / 0 = -inf 0 / 0 = nan inf / 0 = inf -inf / 0 = -inf inf / inf = nan Floating point multiplication: 1 * 0 = 0-1 * 0 = -0 1 * inf = inf -1 * inf = -inf 0 * inf = nan -inf * 0 = nan inf * inf = inf inf * -inf = -inf 1 * nan = nan -1 * nan = nan 0 * nan = nan -0 * nan = nan Floating point addition and subtraction: 1 + inf = inf 1 - inf = -inf inf + inf = inf inf - inf = nan 1 + nan = nan 1 - nan = nan Can you think of a good reason why both 0 and inf should carry sign information? 3. The program bitwise.cpp displays the underlying 32-bit binary patterns that encode the floating point representations of 1, 3, and 18. The header bitconvert.h loads in a special type named convert32_t, which can be interpreted both as a float and as a 32-bit unsigned integer type

2 (uint32_t). The values 3F , , and C are directly assigned to its.i32 data member; the corresponding values 1.0F, 3.0F, and -18.0F are accessed from.f32. C++ uses the convention that numbers beginning with the 0x prefix are interpreted in hexadecimal (base 16). Extend the program so that it also outputs the bit encoding for F and -inf. Those values should be assigned to.i32 using the appropriate hexadecimal value. You should make use of the identity ( = ) = ( ). 512 An infinite value is designated by all 1s in the exponent field and all 0s in the fraction field. $ make bitwise g++ -o bitwise bitwise.cpp -O2 -ansi -pedantic -Wall -lm $./bitwise Single-precision floating point: inf 4. Compile the program max.cpp and pipe the resulting executable to the more command. This will allow you to scroll through the results (page-by-page using the space bar, and line-by-line using the up and down arrow keys). $ make max g++ -o max max.cpp -O2 -ansi -pedantic -Wall -lm $./max more e e e e e e e e e e e e e e e e e e+38 inf e e+39 inf e e+40

3 inf e e+307 inf e e+308 inf inf e+309 inf inf e+310 inf inf e+4931 inf inf e+4932 inf inf inf This program computes the sequence 1, 10, 100, 1000, using floating point numbers of varying width. Convince yourself that the largest finite numerical values of type float, double, and long double are on the order of 10 38, , and , respectively. Notice that there is no loss of precision in the sequence until an element is no longer representable. 5. Create a program file min.cpp using max.cpp as a template. It may be convenient to start with a copy: $ cp max.cpp min.cpp $ emacs min.cpp & The new program should compute the decreasing sequence 1, , using the float, double, and long double types. Have the program indicate exact zeros in text form. 10, 1 100, 1 $ cp max.cpp min.cpp $ emacs min.cpp & $ make min g++ -o min min.cpp -O2 -ansi -pedantic -Wall -lm $./min more e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-45 exact zero e e-46 exact zero e e-47 exact zero e e-308 exact zero e e-309 exact zero e e-310 exact zero e e-311 exact zero e e-312 exact zero e e-313 exact zero e e-314

4 exact zero e e-315 exact zero e e-316 exact zero e e-317 exact zero e e-318 exact zero e e-319 exact zero e e-320 exact zero e e-321 exact zero e e-322 exact zero e e-323 exact zero exact zero e-324 exact zero exact zero e-325 exact zero exact zero e-4935 exact zero exact zero e-4936 exact zero exact zero e-4938 exact zero exact zero e-4938 exact zero exact zero e-4940 exact zero exact zero e-4941 exact zero exact zero e-4942 exact zero exact zero e-4943 exact zero exact zero e-4943 exact zero exact zero e-4945 exact zero exact zero e-4946 exact zero exact zero e-4947 exact zero exact zero e-4948 exact zero exact zero e-4949 exact zero exact zero e-4950 exact zero exact zero e-4950 exact zero exact zero exact zero The approach to zero is quite different from the approach to infinity that we encountered in question 4. Here, there is a steady loss of precision in the steps immediately preceding exact zero. Think about why that it is. 6. Modify the program denorm.cpp so that the convert32_t variable S.F32 = 5.9E-37F is repeatedly halved until all of its bits are equal to zero. This process illustrates how the loss of precision comes about in so-called denormalized numbers. $ make denorm g++ -o denorm denorm.cpp -O2 -ansi -pedantic -Wall -lm $./denorm more Single-precision floating point: e e e-37

5 e e e e e e e e e e e e

6 7. Write a program sum.cpp that computes the partial sum for each of N = 10, 100, 1000,, Accumulate the terms in a float variable in both ascending and descending order, and be sure to store the index n in an integer type wide enough to accommodate the largest upper limit of the sum. The two results should be written in columns alongside the corresponding value of N. The view7.gp script performs a visual comparison with the correct asymptotic behaviour, log N + γ + 1 2N N 2 + Which summation direction is better and why? $ make sum g++ -o sum sum.cpp -O2 -ansi -pedantic -Wall -lm $./sum $ gnuplot -persist view7.gp 8. In various scientific settings, we often need to accumulate numerical values and their powers (e.g., to compute the standard deviation of a large data set). When the values span a wide range of orders of magnitude, the way in which the numbers are summed can make a big difference in how much accuracy is retained. A particularly difficult case occurs when we try to sum non-converging partial series. Let s compare to some exact results: n = N(N + 1), 2 1 n n 2 = N ( 2N 2 + 3N + 1 ), 6 n 3 = N 2 ( N 2 + 2N + 1 ). 4 The values for N = are given in the table below. We discover that the second and third powers can t even be represented exactly within single-precision floating-point scheme! power value of the sum floating-point e e e+19 We will use the seq command (which you ll have to replace with jot in BSD Unix environments) to generate incrementing sequences and awk to raise those numbers to the requisite powers. For example, $ seq 5 awk { print $1, $1*$1, $1*$1*$1 }

7 The program sumdata.cpp reads values from cin as floats and reports their sum based on the functions provided in add_methods.cpp. For everything to work correctly, you ll have to fill in the missing function bodies in that file. The methods provided include summing the terms in increasing and decreasing order, summing the sorted list pairwise (as we discussed in class), summing in conventional order with a high-precision (long double) accumulator variable, and a compensated summation scheme due to Kahan. For those methods that require sorting, make use of the STL sort function provided by the algorithm header. You can read about compensated summation here: When you re done, you should be able to reproduce the following session exactly. $ make sumdata g++ -c -o sumdata.o sumdata.cpp g++ -c -o add_methods.o add_methods.cpp g++ -o sumdata sumdata.o add_methods.o -O2 -ansi -pedantic -Wall -lm $ seq /sumdata I ve read in elements conventional sum: e+09 : increasing sum: e+09 : decreasing sum: e+09 : pairwise sum: e+09 : compensated sum: e+09 : high-prec. sum: e+09 : Note that as many as nine binary digits in the fraction field may disagree depending on the method. The last two methods agree and produce values quite close to the exact result. $ seq awk { print $1*$1 }./sumdata I ve read in elements conventional sum: e+14 : increasing sum: e+14 : decreasing sum: e+14 : pairwise sum: e+14 : compensated sum: e+14 : high-prec. sum: e+14 : $ seq awk { print $1*$1*$1 }./sumdata I ve read in elements conventional sum: e+19 : increasing sum: e+19 : decreasing sum: e+19 : pairwise sum: e+19 : compensated sum: e+19 : high-prec. sum: e+19 :