diff --git a/Makefile b/Makefile index 404699a..f5ef25e 100644 --- a/Makefile +++ b/Makefile @@ -1,46 +1,103 @@ # Name: Makefile -# Description: Builds ./bin/polytest, to test the polyfit() function. +# Description: Builds ./bin/test, to test the polyfit() function. # Test Environment: Ubuntu 18.04.4 LTS, GNU Make 4.1, gcc version 7.4.0. # Author: Henry M. Forson, Melbourne, Florida USA +# Modified by: Stephen P. Morgan, San Jose, California USA # # Synopsis: -# make clean -- removes old build files from the repo. -# make -- builds the ./bin/polytest executable. -# ./bin/polytest -- runs the executable. +# make clean -- removes old build files from the repo. +# make -- builds the ./bin/test-[1-3] executables. +# ./bin/test-1 -- runs the test-1 executable. +# ./bin/test-2 -- runs the test-2 executable. +# ./bin/test-3 -- runs the test-3 executable. +# make test -- build and runs all the test-[1-3] executables # # Results: # $ make clean -# rm -f ./obj/*.o *~ core /*~ ./bin/polytest +# rm -f ./obj/*.o *~ core /*~ | true +# rm -f ./bin/test-1 | true +# rm -f ./bin/test-2 | true +# rm -f ./bin/test-3 | true +# # $ make -# mkdir -p ./obj -# mkdir -p ./bin -# gcc -c -o obj/test.o src/test.c -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -# gcc -c -o obj/polyfit.o src/polyfit.c -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -# gcc -o ./bin/polytest obj/test.o obj/polyfit.o -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -lm -# $ ./bin/polytest -# Test 1 expected (-3.000000 * x) + 5.000000 -# Test 1 produced (-3.000000 * x) + 5.000000 -# Test 1 passed OK. - -# Test 2 expected (-0.142857 * x^2) + 0.485714 -# Test 2 produced (-0.142857 * x^2) + 0.485714 -# Test 2 passed OK. - -# Test 3 expected error = -4 -# Test 3 produced error = -4 -# Test 3 passed OK. - -# Test 4 expected (1.518293 * x) + 0.304878 -# Test 4 produced (1.518293 * x) + 0.304878 -# Test 4 passed OK. - -# Tests complete: 4 passed, 0 failed. - +# gcc -c -o obj/test-1.o src/test-1.c -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror +# gcc -c -o obj/polyfit.o src/polyfit.c -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror +# gcc -o bin/test-1 obj/test-1.o obj/polyfit.o -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror -lm +# gcc -c -o obj/test-2.o src/test-2.c -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror +# gcc -o bin/test-2 obj/test-2.o obj/polyfit.o -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror -lm +# gcc -c -o obj/test-3.o src/test-3.c -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror +# gcc -o bin/test-3 obj/test-3.o obj/polyfit.o -I./inc -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror -lm +# +# $ ./bin/test-1 +# Test 1 expected (-3 * x) + 5 +# Test 1 produced (-3 * x) + 5 +# Test 1 passed OK. +# +# Test 2 expected (-0.142857142857 * x^2) + 0.485714285714 +# Test 2 produced (-0.142857142857 * x^2) + 0.485714285714 +# Test 2 passed OK. +# +# Test 3 expected error = -4 +# Test 3 produced error = -4 +# Test 3 passed OK. +# +# Test 4 expected (1.51829268293 * x) + 0.30487804878 +# Test 4 produced (1.51829268293 * x) + 0.30487804878 +# Test 4 passed OK. +# +# Tests complete: 4 passed, 0 failed. +# +# $ ./bin/test-2 +# Test: Polynomial Order 0: +# Test produced 0.257133485183 +# Error was 0.573962. +# ------------------- +# Test: Polynomial Order 2: +# Test produced (-1.79342717745e-08 * x^2) + (9.56639730699e-06 * x) + 0.259546158319 +# Error was 0.057347; error improvement: total 90.009%, step 90.009%. +# ------------------- +# Test: Polynomial Order 4: +# Test produced (-5.22695213715e-14 * x^4) + (1.69607359765e-10 * x^3) + (-1.96715727627e-07 * x^2) + (7.83400126084e-05 * x) + 0.252368435981 +# Error was 0.016800; error improvement: total 97.073%, step 70.705%. +# ------------------- +# Test: Polynomial Order 6: +# Test produced (-6.10473179655e-19 * x^6) + (2.37734934503e-15 * x^5) + (-3.64187375235e-12 * x^4) + (2.80444481787e-09 * x^3) + (-1.15841078401e-06 * x^2) + (0.00023620315162 * x) + 0.243908991045 +# Error was 0.010339; error improvement: total 98.199%, step 38.456%. +# ------------------- +# Test: Polynomial Order 8: +# Test produced (-5.74676299091e-25 * x^8) + (3.10128708978e-21 * x^7) + (-7.5169670753e-18 * x^6) + (1.05850700619e-14 * x^5) + (-9.2593742705e-12 * x^4) + (5.03106050983e-09 * x^3) + (-1.64535953891e-06 * x^2) + (0.000288160416527 * x) + 0.241921948336 +# Error was 0.010258; error improvement: total 98.213%, step 0.780%. +# +# The order 6 polynomial is the best fit. +# +# $ ./bin/test-3 +# Searching for the optimal number of threads for a system with 56 CPUs. +# +# Test: Polynomial Order 6: +# +# (-6.10473179655e-19 * x^6) + (2.37734934503e-15 * x^5) + (-3.64187375235e-12 * x^4) + (2.80444481787e-09 * x^3) + (-1.15841078401e-06 * x^2) + (0.00023620315162 * x) + 0.243908991045 +# +# The overall fitting error for that polynomial is 0.010339. +# +# Searching for roots of the derivative of that polynomial (for local maxima and minima). +# +# The derivative of that polynomial is (-3.66283907793e-18 * x^5) + (1.18867467252e-14 * x^4) + (-1.45674950094e-11 * x^3) + (8.41333445362e-09 * x^2) + (-2.31682156803e-06 * x) + 0.00023620315162 +# +# polyRoot returned 964 (iterations 1, epsilon 0). +# +# polyEvaluate(964) returned 0.251574. +# +# The optimal number of threads (by CPUs) is 4xCPUs, or 224 (value 0.262309). +# +# The optimal number of threads (individually) is 240 (value 0.262335), or about 4.3xCPUs. +# +# Root 964 is a local minimum. #------------------------------------------------------------------------------------ # MIT License # # Copyright (c) 2020 Henry M. Forson +# Copyright (c) 2021 Stephen P. Morgan # # Permission is hereby granted, free of charge, to any person obtaining a copy # of this software and associated documentation files (the "Software"), to deal @@ -61,41 +118,88 @@ # SOFTWARE. #------------------------------------------------------------------------------------ +default: all IDIR =./inc -CC=gcc -CFLAGS=-I$(IDIR) -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic - ODIR =./obj LDIR =./lib SDIR =./src BDIR =./bin +CC=gcc +CFLAGS=-I$(IDIR) -std=c99 -pedantic-errors -fsanitize=address -fno-omit-frame-pointer -fsanitize=leak -fsanitize=undefined -fsanitize=bounds -fsanitize=null -Wall -Wextra -Wpedantic -Werror + LIBS =-lm -_DEPS = polyfit.h -DEPS = $(patsubst %,$(IDIR)/%,$(_DEPS)) +_TEST-1-DEPS = polyfit.h +TEST-1-DEPS = $(patsubst %,$(IDIR)/%,$(_TEST-1-DEPS)) -_OBJ = test.o polyfit.o -OBJ = $(patsubst %,$(ODIR)/%,$(_OBJ)) +_TEST-1-OBJ = test-1.o polyfit.o +TEST-1-OBJ = $(patsubst %,$(ODIR)/%,$(_TEST-1-OBJ)) -#./obj/test.o: ./src/test.c $(DEPS) -# $(CC) -c -o ./obj/test.o ./src/test.c $(CFLAGS) +$(BDIR)/test-1: $(TEST-1-OBJ) + $(CC) -o $@ $^ $(CFLAGS) $(LIBS) -all: dirs polytest +_TEST-2-DEPS = polyfit.h test-2-x.h test-2-y.h +TEST-2-DEPS = $(DEPS) + $(patsubst %,$(IDIR)/%,$(_TEST-2-DEPS)) -dirs: - mkdir -p $(ODIR) - mkdir -p $(BDIR) +_TEST-2-OBJ = test-2.o polyfit.o +TEST-2-OBJ = $(patsubst %,$(ODIR)/%,$(_TEST-2-OBJ)) + +$(BDIR)/test-2: $(TEST-2-OBJ) + $(CC) -o $@ $^ $(CFLAGS) $(LIBS) + +_TEST-3-DEPS = polyfit.h test-3-x.h test-3-y.h +TEST-3-DEPS = $(DEPS) + $(patsubst %,$(IDIR)/%,$(_TEST-3-DEPS)) + +_TEST-3-OBJ = test-3.o polyfit.o +TEST-3-OBJ = $(patsubst %,$(ODIR)/%,$(_TEST-3-OBJ)) + +$(BDIR)/test-3: $(TEST-3-OBJ) + $(CC) -o $@ $^ $(CFLAGS) $(LIBS) $(ODIR)/%.o: $(SDIR)/%.c $(DEPS) $(CC) -c -o $@ $< $(CFLAGS) -polytest: $(OBJ) - $(CC) -o $(BDIR)/$@ $^ $(CFLAGS) $(LIBS) +.PHONY: all clean dirs tar tarnogit test test-1 test-2 test-3 -.PHONY: clean +all: test-1 test-2 test-3 clean: - rm -f $(ODIR)/*.o *~ core $(INCDIR)/*~ $(BDIR)/polytest + rm -f $(ODIR)/*.o *~ core $(INCDIR)/*~ *.html | true + rm -f $(BDIR)/test-1 | true + rm -f $(BDIR)/test-2 | true + rm -f $(BDIR)/test-3 | true + @if [ -d ${ODIR} ] ; then \ + rmdir --ignore-fail-on-non-empty $(ODIR) ; \ + fi + @if [ -d ${BDIR} ] ; then \ + rmdir --ignore-fail-on-non-empty $(BDIR) ; \ + fi + +dirs: + @if [ ! -d ${ODIR} ] ; then \ + mkdir -p $(ODIR) ; \ + fi + @if [ ! -d ${BDIR} ] ; then \ + mkdir -p $(BDIR) ; \ + fi + +tar: clean + tar czvpf ../polyfit.tgz . + +tarnogit: clean + tar czvpf ../polyfit.tgz --exclude=.git . + + +test: all + $(BDIR)/test-1 + $(BDIR)/test-2 + $(BDIR)/test-3 + +test-1: dirs $(BDIR)/test-1 + +test-2: dirs $(BDIR)/test-2 + +test-3: dirs $(BDIR)/test-3 diff --git a/README.md b/README.md index 581947f..cff596a 100644 --- a/README.md +++ b/README.md @@ -1,53 +1,6 @@ -Name -==== +# Overview -**polyfit** — "C" function that fits a polynomial to a set of points - -Synopsis -======== - -**#include "polyfit.h"** - -__int polyfit( int__ _pointCount_, **double \***_xValues_, **double \***_yValues_, __int__ _coefficientCount_, **double \***_coefficientResults_ __);__ - -Description -=========== - -The **polyfit**() function regresses a line or a higher-order polynomial to a set of points, using the Method of Least Squares. Its design goals were simplicity and ease of porting, as opposed to run-time efficiency with large data. - -Arguments ---------- -_pointCount_ — input. The total number of points in the set. For the algorithm to work, this must be greater than or equal to _coefficientCount_. - -_xValues_ — input. Points to an array of the X coordinates of the points. There should be _pointCount_ X coordinates. - -_yValues_ — input. Points to an array of the Y coordinates of the points. There should be _pointCount_ Y coordinates. - -_coefficientCount_ — input. The number of coefficients to be computed, equal to the degree of the polynomial plus 1. For instance, if fitting a line — a first degree polynomial — then _coefficientCount_ would be 2, and for fitting a parabola — a second degree polynomial — _coefficientCount_ would be 3. - -_results_ — input. Points to where the computed coefficients will be stored. There should be space for _coefficientCount_ coefficients. These coefficients are ordered from the highest-order monomial term to the lowest, such that for instance the polynomial: - - 5x² + 3x - 7 - -is represented as: - - [ 5, 3, -7 ] - - -Return Value ------------- -In addition to setting the coefficient results, the **polyfit**() function returns **0** on success. - -FILES ------ -*./src/polyfit.c* — defines the **polyfit**() function. - -*./inc/polyfit.h* — declares the **polyfit**() function's prototype. - -MISC ----- -To support unit testing on Linux with gcc, the repo has these additional files: - -*./src/test.c* — exercises **polyfit**() and provides examples of usage. - -*./Makefile* — allows the *make* command to build an executable, *./bin/polytest*, that tests **polyfit**(). +This document describes the coding interface for the Polyfit package of +polynomial curve fitting tools. The package includes: [`polyfit`](polyfit.md), +[`polyDerivative`](polyDerivative.md), [`polyError`](polyError.md), +[`polyEvaluate`](polyEvaluate.md), and [`polyRoot`](polyRoot.md). diff --git a/inc/polyfit.h b/inc/polyfit.h index c2f9b74..a3ed4ff 100644 --- a/inc/polyfit.h +++ b/inc/polyfit.h @@ -42,7 +42,7 @@ // // Returns 0 if success. //-------------------------------------------------------- -int polyfit( int pointCount, double *xValues, double *yValues, int coefficientCount, double *coefficientResults ); +int polyfit(int pointCount, const double *xValues, const double *yValues, int coefficientCount, double *coefficientResults); //-------------------------------------------------------- // polyToString() @@ -50,8 +50,14 @@ int polyfit( int pointCount, double *xValues, double *yValues, int coefficientCo // its coefficients. // Returns 0 on success. //-------------------------------------------------------- -int polyToString( char *stringBuffer, size_t stringBufferSz, int coeffCount, double *coefficients ); +int polyToString(char *stringBuffer, size_t stringBufferSz, int coeffCount, double *coefficients); +void polyDerivative(int cc, const double* cr_in, double* cr_out); +double polyError(int pc, const double* x, const double* y, int cc, const double* cr); + +double polyEvaluate(int cc, const double* cr, double x); + +double polyRoot(int cc, const double* cr, double x0, double x1, int* iterations, double* epsilon); #endif // POLYFIT_H diff --git a/inc/test-2-x.h b/inc/test-2-x.h new file mode 100644 index 0000000..215596d --- /dev/null +++ b/inc/test-2-x.h @@ -0,0 +1,32396 @@ + 56, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 57, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 58, + 59, + 59, + 59, 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+0.249581, +0.249202, +0.250907, +0.251095, diff --git a/inc/test-3-x.h b/inc/test-3-x.h new file mode 120000 index 0000000..67a8fb1 --- /dev/null +++ b/inc/test-3-x.h @@ -0,0 +1 @@ +test-2-x.h \ No newline at end of file diff --git a/inc/test-3-y.h b/inc/test-3-y.h new file mode 120000 index 0000000..8a41d17 --- /dev/null +++ b/inc/test-3-y.h @@ -0,0 +1 @@ +test-2-y.h \ No newline at end of file diff --git a/polyDerivative.md b/polyDerivative.md new file mode 100644 index 0000000..ff2acad --- /dev/null +++ b/polyDerivative.md @@ -0,0 +1,49 @@ +Name +==== + +**polyDerivative** — "C" function that computes a derivative of a polynomial. + +Synopsis +======== + +**#include "polyfit.h"** + +__void polyDerivative(int__ _coefficientCount_, **const double\*** _coefficientResultsIn_, **double\*** _coefficientResultsOut___);__ + +Description +=========== + +The **polyDerivative**() function computes the derivative of a polynomial. + +Arguments +--------- + +_coefficientCount_ — input. The number of coefficients to be input, equal to the degree of the polynomial plus 1. + +_coefficientResultsIn_ — input. The coefficients of the polynomial. There should be _coefficientCount_ coefficients. +These coefficients are ordered from the highest-order monomial term to the lowest, such that for instance the polynomial: +5x² + 3x - 7 is represented as: [ 5, 3, -7 ] + +_coefficientResultsOut_ — ouput. The coefficients of the derivative of the polynomial. +Note that one less coefficient is output than input. These are stored in the same order as _coefficientResultsIn_. +Also, space for the output coeffcient array must be allocated by the caller. + +Return Value +------------ +No value is returned. + +FILES +----- +*./src/polyfit.c* — defines the **polyDerivative**() function. + +*./inc/polyfit.h* — declares the **polyDerivative**() function's prototype. + +MISC +---- +To support unit testing on Linux with gcc, the repo has these additional files: + +*./src/test-2.c* — exercises **polyDerivative**() and provides examples of usage. + +*./src/test-3.c* — exercises **polyDerivative**() and provides examples of usage. + +*./Makefile* — allows the *make* command to build the executables, *./bin/test-2* and *./bin/test-3* that test **polyDerivative**(). diff --git a/polyError.md b/polyError.md new file mode 100644 index 0000000..8755d25 --- /dev/null +++ b/polyError.md @@ -0,0 +1,48 @@ +Name +==== + +**polyError** — "C" function that estimates the error of a polynomial fitted via **polyfit**() vs. the data used to fit the polynomial. + +Synopsis +======== + +**#include "polyfit.h"** + +__double polyError(int__ _pointCount_, **const double\*** _xValues_, **const double\*** _yValues_, __int__ _coefficientCount_, **const double\*** _coefficients___);__ + +Description +=========== + +The **polyError**() function estimates the error of a polynomial fitted via **polyfit**() vs. the data used to fit the polynomial. + +Arguments +--------- +_pointCount_ — input. The total number of points in the set. This must be equal to _coefficientCount_. + +_xValues_ — input. Points to an array of the X coordinates of the points. There should be _pointCount_ X coordinates. + +_yValues_ — input. Points to an array of the Y coordinates of the points. There should be _pointCount_ Y coordinates. + +_coefficientCount_ — input. The number of coefficients, equal to the degree of the polynomial plus 1. + +_coefficients_ — input. Points to where the computed coefficients are stored. There should be space for _coefficientCount_ coefficients. These coefficients are ordered from the highest-order monomial term to the lowest, such that for instance the polynomial: 5x² + 3x - 7 is represented as: [ 5, 3, -7 ]. + +Return Value +------------ +The **polyError**() function returns the sum of the errors (squared). + +FILES +----- +*./src/polyfit.c* — defines the **polyError**() function. + +*./inc/polyfit.h* — declares the **polyError**() function's prototype. + +MISC +---- +To support unit testing on Linux with gcc, the repo has these additional files: + +*./src/test-2.c* — exercises **polyError**() and provides examples of usage. + +*./src/test-3.c* — exercises **polyError**() and provides examples of usage. + +*./Makefile* — allows the *make* command to build executables *./bin/test-2* and *./bin/test-3* that test **polyError**(). diff --git a/polyEvaluate.md b/polyEvaluate.md new file mode 100644 index 0000000..ea9efeb --- /dev/null +++ b/polyEvaluate.md @@ -0,0 +1,44 @@ +Name +==== + +**polyEvaluate** — "C" function that evaluates a polynomial at a given _x_ value. + +Synopsis +======== + +**#include "polyfit.h"** + +__double polyEvaluate(int__ _coefficientCount_, **const double\*** _coefficients_, **double** _x___);__ + +Description +=========== + +The **polyEvaluate**() function evaluates a polynomial at a given _x_ value. + +Arguments +--------- +_coefficientCount_ — input. The number of coefficients of the polynomial, equal to the degree of the polynomial plus 1. + +_coefficients_ — input. Points to where the computed coefficients are stored. There should be _coefficientCount_ coefficients. +These coefficients are ordered from the highest-order monomial term to the lowest, such that for instance the polynomial: +5x² + 3x - 7 is represented as: [ 5, 3, -7 ]. + +_x_ - input. The _x_ value at which the polynomial is to be evaluated. + +Return Value +------------ +The **polyEvaluate**() function returns the value of the polynomial at _x_. + +FILES +----- +*./src/polyfit.c* — defines the **polyEvaluate**() function. + +*./inc/polyfit.h* — declares the **polyEvaluate**() function's prototype. + +MISC +---- +To support unit testing on Linux with gcc, the repo has these additional files: + +*./src/test-2.c* and *.src.test-3.c* — exercise **polyEvaluate**() and provide examples of usage. + +*./Makefile* — allows the *make* command to build executables *./bin/test-2* and *./bin/test-3* that test **polyEvaluate**(). diff --git a/polyRoot.md b/polyRoot.md new file mode 100644 index 0000000..516b318 --- /dev/null +++ b/polyRoot.md @@ -0,0 +1,62 @@ +Name +==== + +**polyRoot** — "C" function that computes a real root of a polynomial within the range [_x0_, _x1_]. + +Synopsis +======== + +**#include "polyfit.h"** + +__double polyRoot(int__ _coefficientCount_, **const double\*** _coefficients_ **double** _x0_, **double** _x1_, __int\*__ _iterations_, **double\*** _epsilon___);__ + +Description +=========== + +The **polyRoot**() function computes a real root of a polynomial within the range [_x0_, _x1_], +in no more than _\*iterations_ and with no more then _\*epsilon_ error. +Note that, if _\*iterations_ returns with a larger value than input, the method did not meet +its requirements, and the error of the approximation is _\*epsilon_. + +**polyRoot**() uses the well-known _secant method_ of root approximation. + +Arguments +--------- +_coefficientCount_ — input. The number of coefficients.i, equal to the degree of the polynomial plus 1. +For instance, if the polynomial is of the first degree, i.e., a line, then _coefficientCount_ would be 2, and if of of the second degree, i.e., a parabola, then _coefficientCount_ would be 3. + +_coefficients_ — input. Points to where the computed coefficients are stored. There should be _coefficientCount_ coefficients. +These coefficients are ordered from the highest-order monomial term to the lowest, such that for instance the polynomial: +5x² + 3x - 7 is represented as: [ 5, 3, -7 ]. + +_x0_ — input. The value of the start of the range. + +_x1_ — input. The value of the end of the range. + +_iterations_ — input/output. On input, points to the value of the maximum number of iterations the method should employ. +On output, points to the value of the number of iterations actually employed. +Note: On input, _iterations_ may be _NULL_, in which case the method will employ no more than 16 iterations. + +_epsilon_ — input/output. On input, points to the value of the maximum error that may exist between the root and the estimate returned. +On output, points to the value of the actual error determined. +Note: On input, _epsilon_ may be _NULL_, in which case the method will seek an error of no more than 0.000001. + +Return Value +------------ +The **polyRoot**() function returns the real root on success. + +FILES +----- +*./src/polyfit.c* — defines the **polyRoot**() function. + +*./inc/polyfit.h* — declares the **polyRoot**() function's prototype. + +MISC +---- +To support unit testing on Linux with gcc, the repo has these additional files: + +*./src/test-2.c* — exercises **polyRoot**() and provides examples of usage. + +*./src/test-3.c* — exercises **polyRoot**() and provides examples of usage. + +*./Makefile* — allows the *make* command to build executables *./bin/test-2* and *./bin/test-3* that test **polyRoot**(). diff --git a/polyfit.md b/polyfit.md new file mode 100644 index 0000000..8631b80 --- /dev/null +++ b/polyfit.md @@ -0,0 +1,46 @@ +Name +==== + +**polyfit** — "C" function that fits a polynomial to a set of points + +Synopsis +======== + +**#include "polyfit.h"** + +__int polyfit(int__ _pointCount_, **const double\*** _xValues_, **const double\*** _yValues_, __int__ _coefficientCount_, **double\*** _coefficientResults___);__ + +Description +=========== + +The **polyfit**() function regresses a line or a higher-order polynomial to a set of points, using the Method of Least Squares. Its design goals were simplicity and ease of porting, as opposed to run-time efficiency with large data. + +Arguments +--------- +_pointCount_ — input. The total number of points in the set. For the algorithm to work, this must be greater than or equal to _coefficientCount_. + +_xValues_ — input. Points to an array of the X coordinates of the points. There should be _pointCount_ X coordinates. + +_yValues_ — input. Points to an array of the Y coordinates of the points. There should be _pointCount_ Y coordinates. + +_coefficientCount_ — input. The number of coefficients to be computed, equal to the degree of the polynomial plus 1. For instance, if fitting a line — a first degree polynomial — then _coefficientCount_ would be 2, and for fitting a parabola — a second degree polynomial — _coefficientCount_ would be 3. + +_results_ — input. Points to where the computed coefficients will be stored. There should be space for _coefficientCount_ coefficients. These coefficients are ordered from the highest-order monomial term to the lowest, such that for instance the polynomial: 5x² + 3x - 7 is represented as: [ 5, 3, -7 ]. + +Return Value +------------ +In addition to setting the coefficient results, the **polyfit**() function returns **0** on success. + +FILES +----- +*./src/polyfit.c* — defines the **polyfit**() function. + +*./inc/polyfit.h* — declares the **polyfit**() function's prototype. + +MISC +---- +To support unit testing on Linux with gcc, the repo has these additional files: + +*./src/test-1.c* — exercises **polyfit**() and provides examples of usage. + +*./Makefile* — allows the *make* command to build an executable, *./bin/test-1*, that tests **polyfit**(). diff --git a/src/polyfit.c b/src/polyfit.c index e44df28..39d03ff 100644 --- a/src/polyfit.c +++ b/src/polyfit.c @@ -6,6 +6,7 @@ // MIT License // // Copyright (c) 2020 Henry M. Forson +// Copyright (c) 2021 Stephen P. Morgan // // Permission is hereby granted, free of charge, to any person obtaining a copy // of this software and associated documentation files (the "Software"), to deal @@ -26,8 +27,11 @@ // SOFTWARE. //------------------------------------------------------------------------------------ +#include #include // pow() #include // bool +#include +#include #include // printf() #include // calloc() #include // strlen() @@ -102,7 +106,7 @@ static matrix_t * createProduct( matrix_t *pLeft, matrix_t *pRight ); // -4 if unable to solve equations. //-------------------------------------------------------- //int polyfit( int pointCount, point_t pointArray[], int coeffCount, double coeffArray[] ) -int polyfit( int pointCount, double *xValues, double *yValues, int coefficientCount, double *coefficientResults ) +int polyfit( int pointCount, const double *xValues, const double *yValues, int coefficientCount, double *coefficientResults ) { int rVal = 0; int degree = coefficientCount - 1; @@ -280,15 +284,15 @@ int polyToString( char *stringBuffer, size_t stringBufferSz, int coeffCount, dou if( 0 == exponent ) { - snprintf( pNext, remainingSize, "%s%f", isThisTheFirstTermShown ? "" : " + ", coefficients[ i ] ); + snprintf( pNext, remainingSize, "%s%.12g", isThisTheFirstTermShown ? "" : " + ", coefficients[ i ] ); } else if( 1 == exponent) { - snprintf( pNext, remainingSize, "%s(%f * x)", isThisTheFirstTermShown ? "" : " + ", coefficients[ i ] ); + snprintf( pNext, remainingSize, "%s(%.12g * x)", isThisTheFirstTermShown ? "" : " + ", coefficients[ i ] ); } else { - snprintf( pNext, remainingSize, "%s(%f * x^%d)", isThisTheFirstTermShown ? "" : " + ", coefficients[i], exponent ); + snprintf( pNext, remainingSize, "%s(%.12g * x^%d)", isThisTheFirstTermShown ? "" : " + ", coefficients[i], exponent ); } isThisTheFirstTermShown = false; } @@ -296,6 +300,143 @@ int polyToString( char *stringBuffer, size_t stringBufferSz, int coeffCount, dou return 0; } +//-------------------------------------------------------- +// polyMultiply() +// Returns a value of x multiplied by itself n times, e.g., +// x^2 or x^3. +//-------------------------------------------------------- +static inline double +polyMultiply(double x, int n) +{ + if (n == 0) { + return 1.0; + } + else { + return x * polyMultiply(x, n - 1); + } +} + +//-------------------------------------------------------- +// polyError() +// Evaluates the error of a polynomial vs. the original +// data used to regress the polynomial, at a point . +// Retiurns the error (squared) at that point. +//-------------------------------------------------------- +double +polyError(int pc, const double* x, const double* y, int cc, const double* cr) +{ + double error = 0.0; + + for (int n = 0; n < pc; n++) { + double x_val = x[n]; + double y_val = y[n]; + + double y_est = 0.0; + + for (int i = 0; i < cc; i++) { + y_est += polyMultiply(x_val, cc - i - 1) * *(cr + i); + } + + double y_trm = y_est - y_val; + + error += y_trm * y_trm; + } + + return error; +} + +//-------------------------------------------------------- +// polyEvaluate() +// Evaluates a polynomial at x. +// Returns the value of the polynomial at x. +//-------------------------------------------------------- +double +polyEvaluate(int cc, const double* cr, double x) +{ + double value = 0.0; + + for (int i = 0; i < cc; i++) { + double coeff = *(cr + i); + value += coeff * polyMultiply(x, cc - i - 1); + } + + return value; +} + + +//-------------------------------------------------------- +// polyRoot() +// +// Determins a real root of a given polynomial in the range +// [x0..x1]. Iterates no more than *iteration times or until the +// difference between values in the iteration is less than *epsilon. +// Uses the secant iterative root-finding method. If epsilon is +// NULL, uses an epsilon valu eof 0.000001; otherwise, returns the +// actual epsilon found. If iterations is NULL, uses an iterations +// value of 16; otherwise, returns the actual number of iterations. +// +// Returns the vaue of one real root in the range (if any). Note +// that, if *epsilon or *iterations is hit, the value returned will +// not be valid. Thus, those values should be checked on exit. +// Call this function with epsilon and/or iterations set to NULL +// at your own risk. +//-------------------------------------------------------- + +// Value to limit number of iterations in polyRoot. + +double +polyRoot(int cc, const double* cr, double x0, double x1, + int* iterations, double* epsilon) +{ + double x0_init = x0; + double x1_init = x1; + double my_epsilon = (epsilon == NULL) ? 0.000001 : *epsilon; + int my_iterations = (iterations == NULL) ? 16 : *iterations; + int count; + + double x2 = (x0 + x1) / 2.0; + + for (int i = 0; i < my_iterations; i++) { + x2 = x1 - polyEvaluate(cc, cr, x1) * (x1 - x0) / + (polyEvaluate(cc, cr, x1) - polyEvaluate(cc, cr, x0)); + + x0 = x1; + x1 = x2; + + if (x2 < x0_init || x2 > x1_init || + fabs(x0 - x2) < my_epsilon || fabs(x1 - x2) < my_epsilon) { + count = i + 1; + break; + } + } + + if (epsilon != NULL) { + *epsilon = fabs(x0 - x2) < fabs(x1 - x2) ? + fabs(x0 - x2) : fabs(x1 - x2); + } + + if (iterations != NULL) { + *iterations = count; + } + + return x2; +} + +//-------------------------------------------------------- +// polyDerivative(). +// Computes the derivative of the given polynomial. +// Returns the derivative polynomial in cr_out. The +// derivative polynomial will have (cc - 1) terms. +// Note that cr_out must be supplied by the caller. +//-------------------------------------------------------- +void +polyDerivative(int cc, const double* cr_in, double* cr_out) +{ + for (int i = 0; i < cc - 1; i++) { + *(cr_out + i) = *(cr_in + i) * (double)(cc - i - 1); + } +} + //========================================================= // Private function definitions //========================================================= diff --git a/src/test-1.c b/src/test-1.c new file mode 100644 index 0000000..943b4fd --- /dev/null +++ b/src/test-1.c @@ -0,0 +1,182 @@ +// Name: test.c +// Description: Test file to exercise the MLS polynomial fitting function. +// Author: Henry Forson, Melbourne, Florida USA + +//------------------------------------------------------------------------------------ +// MIT License +// +// Copyright (c) 2020 Henry M. Forson +// +// Permission is hereby granted, free of charge, to any person obtaining a copy +// of this software and associated documentation files (the "Software"), to deal +// in the Software without restriction, including without limitation the rights +// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the Software is +// furnished to do so, subject to the following conditions: +// +// The above copyright notice and this permission notice shall be included in all +// copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +// SOFTWARE. +//------------------------------------------------------------------------------------ + +#include +#include +#include "polyfit.h" + +// Buffer to hold a string representation of a polynomial: +#define POLY_STRING_BF_SZ (256) +char polyStringBf[POLY_STRING_BF_SZ]; + +// ---------------- TEST 1 DATA ------------------------ +// Create test data to find the closest line to (0,6), (1,0), and (2,0). +// See also page 218 of: https://math.mit.edu/~gs/linearalgebra/ila0403.pdf +double x1[] = { 0, 1, 2}; +double y1[] = { 6, 0, 0}; +int pc1 = (int) (sizeof(x1) / sizeof(x1[0])); // pointCount +double cr1[] = {0, 0}; // coefficientResults +int cc1 = (int) (sizeof(cr1) / sizeof(cr1[0])); // coefficientCount +char *er1 = "(-3 * x) + 5"; // expected result + +// ---------------- TEST 2 DATA ------------------------ +// Create test data to find the closest parabola to the points shown. +// as example "4.3 B" on page 226 of: https://math.mit.edu/~gs/linearalgebra/ila0403.pdf +double x2[] = { -2, -1, 00, 1, 2}; +double y2[] = { 0, 0, 1, 0, 0}; +int pc2 = (int) (sizeof(x2) / sizeof(x2[0])); // pointCount +double cr2[] = {0, 0, 0}; // coefficientResults +int cc2 = (int) (sizeof(cr2) / sizeof(cr2[0])); // coefficientCount +char *er2 = "(-0.142857142857 * x^2) + 0.485714285714"; // expected result + +// ---------------- TEST 3 DATA ------------------------ +// Create a failure test case with impossible to solve data. +double x3[] = { 0, 1, 1, 0}; +double y3[] = { 0, 1, 1, 0}; +int pc3 = (int) (sizeof(x3) / sizeof(x3[0])); // pointCount +double cr3[] = {0, 0, 0}; // coefficientResults +int cc3 = (int) (sizeof(cr3) / sizeof(cr3[0])); // coefficientCount +char *er3 = "error = -4"; // expected result + +// ---------------- TEST 4 DATA ------------------------ +// test MLS regeression example from https://www.mathsisfun.com/data/least-squares-regression.html +double x4[] = { 2, 3, 5, 7, 9}; +double y4[] = { 4, 5, 7, 10, 15}; +int pc4 = (int) (sizeof(x4) / sizeof(x4[0])); // pointCount +double cr4[] = {0, 0}; // coefficientResults +int cc4 = (int) (sizeof(cr4) / sizeof(cr4[0])); // coefficientCount +char *er4 = "(1.51829268293 * x) + 0.30487804878"; // expected result + + +//-------------------------------------------------------- +// main() +// Unit tests the poly() function. +//-------------------------------------------------------- +int main() +{ + int rVal; + int passedCount = 0; + int failedCount = 0; + + //---------------------TEST 1--------------------------- + printf( "Test 1 expected %s\n", er1); + rVal = polyfit( pc1, x1, y1, cc1, cr1); + if( 0 == rVal) + { + polyToString( polyStringBf, POLY_STRING_BF_SZ, cc1, cr1 ); + } + else + { + snprintf( polyStringBf, POLY_STRING_BF_SZ, "error = %d", rVal ); + } + printf( "Test 1 produced %s\n", polyStringBf); + if( 0 == strcmp( polyStringBf, er1) ) + { + printf( "Test 1 passed OK.\n\n"); + passedCount += 1; + } + else + { + printf( "Test failed.\n\n"); + failedCount += 1; + } + + //---------------------TEST 2--------------------------- + printf( "Test 2 expected %s\n", er2); + rVal = polyfit( pc2, x2, y2, cc2, cr2); + if( 0 == rVal) + { + polyToString( polyStringBf, POLY_STRING_BF_SZ, cc2, cr2 ); + } + else + { + snprintf( polyStringBf, POLY_STRING_BF_SZ, "error = %d", rVal ); + } + printf( "Test 2 produced %s\n", polyStringBf); + if( 0 == strcmp( polyStringBf, er2) ) + { + printf( "Test 2 passed OK.\n\n"); + passedCount += 1; + } + else + { + printf( "Test failed.\n\n"); + failedCount += 1; + } + + //---------------------TEST 3--------------------------- + printf( "Test 3 expected %s\n", er3); + rVal = polyfit( pc3, x3, y3, cc3, cr3); + if( 0 == rVal) + { + polyToString( polyStringBf, POLY_STRING_BF_SZ, cc3, cr3 ); + } + else + { + snprintf( polyStringBf, POLY_STRING_BF_SZ, "error = %d", rVal ); + } + printf( "Test 3 produced %s\n", polyStringBf); + if( 0 == strcmp( polyStringBf, er3) ) + { + printf( "Test 3 passed OK.\n\n"); + passedCount += 1; + } + else + { + printf( "Test failed.\n\n"); + failedCount += 1; + } + +//---------------------TEST 4--------------------------- + printf( "Test 4 expected %s\n", er4); + rVal = polyfit( pc4, x4, y4, cc4, cr4); + if( 0 == rVal) + { + polyToString( polyStringBf, POLY_STRING_BF_SZ, cc4, cr4 ); + } + else + { + snprintf( polyStringBf, POLY_STRING_BF_SZ, "error = %d", rVal ); + } + printf( "Test 4 produced %s\n", polyStringBf); + if( 0 == strcmp( polyStringBf, er4) ) + { + printf( "Test 4 passed OK.\n\n"); + passedCount += 1; + } + else + { + printf( "Test failed.\n\n"); + failedCount += 1; + } + +//---------------------SUMMARY--------------------------- + printf( "Tests complete: %d passed, %d failed.\n", passedCount, failedCount); + return( -failedCount ); +} + diff --git a/src/test-2.c b/src/test-2.c new file mode 100644 index 0000000..7d602dd --- /dev/null +++ b/src/test-2.c @@ -0,0 +1,167 @@ +// Name: test.c +// Description: Test file to exercise the MLS polynomial fitting function. +// Author: Stephen P. Morgan, San Jose, CA +// Derived from work by: Henry Forson, Melbourne, Florida USA + +//------------------------------------------------------------------------------------ +// MIT License +// +// Copyright (c) 2021 Stephen P. Morgan +// +// Permission is hereby granted, free of charge, to any person obtaining a copy +// of this software and associated documentation files (the "Software"), to deal +// in the Software without restriction, including without limitation the rights +// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the Software is +// furnished to do so, subject to the following conditions: +// +// The above copyright notice and this permission notice shall be included in all +// copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +// SOFTWARE. +//------------------------------------------------------------------------------------ + +#include +#include +#include +#include "polyfit.h" + +// Buffer to hold a string representation of a polynomial: + +#define POLY_STRING_BF_SZ (256) + +char polyStringBf[POLY_STRING_BF_SZ]; + +double x[] = { +#include "test-2-x.h" +}; + +double y[] = { +#include "test-2-y.h" +}; + +int cc = 13; + +int pc = (int)(sizeof(x) / sizeof(x[0])); // pointCount + +double* cr; + +//-------------------------------------------------------- +// main() +// +// This is a test program of (augmented) polyfilt which takes +// measurement data "test-2-x.h" and "test-2-y.h", the X and Y +// coordinates of samples taken for the number of threads and +// weighted system performance, respectively, for an actual +// computing system with 56 CPUs. X ranges from [56..1120], or one +// thread per CPU to 20 threads per CPU, and Y ranges from +// [0.0..1.0], where a higher weight corresponds to better +// performance. +// +// The goal of this test is to identify a polynomial regression in +// one variable (X) which optimally models the relationship between +// the number of threads and system performance. +// +// The test works by fitting the data to all order polynomials from +// zeroth to twelfth, and showing the error associated with each. +// The user must "eyeball" the optimal polynomial (if any). +//-------------------------------------------------------- +#define MAX_POLYNOMIAL 12 + +int +main(int argc, char* argv[]) +{ + (void)argc; + (void)argv; + + double error_original; + double error_previous; + + // Generate even-ordered polynomials of orders 0 to 12 to fit the supplied data. + + for (int cc = 1; cc <= MAX_POLYNOMIAL + 1; cc += 2) { + printf("Test: Polynomial Order %d:\n", cc - 1); + + // Allocate storage to hold polynomial coefficients, + + cr = calloc(cc, sizeof(double)); // coefficientResults + + // Fit a polynomial of order cc-1 to the data. + + int rVal = polyfit(pc, x, y, cc, cr); + + // Check whether polyfit was able to generate the desired polynomial. + + if (0 == rVal) { + // Format the polynomial. + + polyToString(polyStringBf, POLY_STRING_BF_SZ, cc, cr); + } + else { + // Format the polynomial (such as it is) with an error message. + + snprintf(polyStringBf, POLY_STRING_BF_SZ, "error = %d", rVal); + } + + // Print polynomial or error string. + + printf("Test produced %s\n", polyStringBf); + + // If no error, analyze the polynomial. + + if (0 == rVal) { + // Compute the error associated with the polynomial. + + double error = polyError(pc, x, y, cc, cr); + + printf("Error was %.6f", error); + + if (cc == 1) { + printf(".\n"); + + // If this is the first valid polynomial, remember its error. + + error_original = error; + } + else { + // If this is not the first valid polynomial, print the change + // in the error overall and for this step. + + printf("; error improvement: total %7.3f%%, step %7.3f%%.\n", + 100.0 * (1.0 - error / error_original), + 100.0 * (1.0 - error / error_previous)); + + if (100.0 * (1.0 - error / error_previous) < 1.0) { + printf("\nThe order %u polynomial is the best fit.\n", cc - 3); + free(cr); + goto Exit; + } + } + + // Remember this error for stepwise error comparison(s). + + error_previous = error; + } + + printf("-------------------\n"); + + // Free allocated resources. + + free(cr); + } + + printf("\nThe %uth-order polynomial is the best fit.\n", MAX_POLYNOMIAL); + +Exit: + + // Done. + + return 0; +} + diff --git a/src/test-3.c b/src/test-3.c new file mode 100644 index 0000000..8efa517 --- /dev/null +++ b/src/test-3.c @@ -0,0 +1,188 @@ +// Name: test-3.c +// Description: Another test file to exercise the MLS polynomial fitting function. +// Author: Stephen P. Morgan, San Jose, CA +// Derived from work by: Henry Forson, Melbourne, Florida USA + +//----------------------------------------------------------------------------- +// MIT License +// +// Copyright (c) 2021 Stephen P. Morgan +// +// Permission is hereby granted, free of charge, to any person obtaining a copy +// of this software and associated documentation files (the "Software"), to deal +// in the Software without restriction, including without limitation the rights +// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the Software is +// furnished to do so, subject to the following conditions: +// +// The above copyright notice and this permission notice shall be included in all +// copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +// SOFTWARE. +//------------------------------------------------------------------------------ + +#include +#include +#include +#include "polyfit.h" + +// Buffer to hold a string representation of a polynomial: + +#define POLY_STRING_BF_SZ (256) + +char polyStringBf[POLY_STRING_BF_SZ]; + +double x[] = { +#include "test-3-x.h" +}; + +double y[] = { +#include "test-3-y.h" +}; + +int cc = 13; + +int pc = (int)(sizeof(x) / sizeof(x[0])); // pointCount + +double* cr; +double* cr_der; + +#define NUM_CPUS 56 + +//-------------------------------------------------------- +// main() +// +// This is a test program of (augmented) polyfilt which takes +// measurement data "test-3-x.h" and "test-3-y.h", the X and Y +// coordinates of samples taken for the number of threads and +// weighted system performance, respectively, for an actual +// computing system with 56 CPUs. X ranges from [56..1120], or one +// thread per CPU to 20 threads per CPU, and Y ranges from +// [0.0..1.0], where a higher weight corresponds to better +// performance. The goal of this test is to identify the optimal +// number of threads with which to run the system, both by the +// number of theads per CPU and the number of threads individually, +// in case they differ. +// +// The test works by fitting the data to a sixth-order polynomial +// (which we've already decided best models the sysem), then +// searching for a root of the derivative of the polynomial (a +// well-known method to find local maxima and minima) within a +// range. (To do so, we implemented the secant method.) +// +// Next, the program searches for the actual optimal configuration +// by evaluating the polynomial at each of the configurations of +// threads ranging from 1xCPUs, 2xCPUs, ..., 20xCPUs. It also +// searches for the actual optimal number of threads by evaluating +// the polynomial at each individual number of threads, from +// [56..1120]. + +// Finally, the program determines whether the root found using +// the secant method represents a local maximum or minimum. +// +// Note: We happen to know, a priori, that the actual optimal number +// of threads lies somewhere in the range [4xCPUs..5xCPUs]. +//-------------------------------------------------------- +int +main(int argc, char* argv[]) +{ + (void)argc; + (void)argv; + + printf("Searching for the optimal number of threads for a system with %u CPUs.\n", + NUM_CPUS); + + int cc = 7; + + printf("\nTest: Polynomial Order %d:\n", cc - 1); + + cr = calloc(cc, sizeof(double)); // coefficientResults + cr_der = calloc(cc - 1, sizeof(double)); // coefficientResults + + int rVal = polyfit(pc, x, y, cc, cr); + + if (0 == rVal) { + polyToString(polyStringBf, POLY_STRING_BF_SZ, cc, cr); + } + else { + snprintf(polyStringBf, POLY_STRING_BF_SZ, "error = %d", rVal); + } + + printf("\n%s\n", polyStringBf); + + printf("\nThe overall fitting error for that polynomial is %.6f.\n", + polyError(pc, x, y, cc, cr)); + + printf("\nSearching for roots of the derivative of that polynomial" + " (for local maxima and minima).\n"); + + polyDerivative(cc, cr, cr_der); + + polyToString(polyStringBf, POLY_STRING_BF_SZ, cc - 1, cr_der); + + printf("\nThe derivative of that polynomial is %s\n", polyStringBf); + + int iterations = 16; + double epsilon = 0.000001; + + int root = polyRoot(cc - 1, cr_der, NUM_CPUS, 20 * NUM_CPUS, + &iterations, &epsilon); + + printf("\npolyRoot returned %u (iterations %u, epsilon %.g).\n", root, + iterations, epsilon); + + double value = polyEvaluate(cc, cr, root); + double max = value; + double root_value = value; + int max_idx = 0; + + printf("\npolyEvaluate(%u) returned %g.\n", root, root_value); + + for (int i = 1; i <= 20; i++) { + value = polyEvaluate(cc, cr, i * NUM_CPUS); + if (value > max) { + max_idx = i; + max = value; + } + } + + double max_by_cpus = max; + + printf("\nThe optimal number of threads (by CPUs) is %uxCPUs, or %u (value %g).\n", + max_idx, max_idx * NUM_CPUS, max_by_cpus); + + for (int i = 1; i <= 1120; i++) { + value = polyEvaluate(cc, cr, i); + if (value > max) { + max_idx = i; + max = value; + } + } + + double max_individual = max; + + printf("\nThe optimal number of threads (individually) is %u (value %g), or about %.1fxCPUs.\n", + max_idx, max_individual, (double)max_idx / NUM_CPUS); + + double max_overall = + (max_by_cpus > max_individual) ? max_by_cpus : max_individual; + + if (max_overall >= root_value) { + printf("\nRoot %u is a local minimum.\n", root); + } + else { + printf("\nRoot is a local maximum.\n"); + } + + free(cr); + free(cr_der); + + return 0; +} +