（免责声明：我对数学非常糟糕，并且来自JavaScript，所以我为任何不准确之处道歉，并会尽力纠正它们。）
Rosetta代码上的example显示了如何使用gsl计算系数。这是代码：
polifitgsl.h：
$('.addName1b').click(function() {
var $this = $(this);
$this.toggleClass('addName1b');
if ($this.hasClass('addName1b')) {
$this.text('Add name');
} else {
$this.text('Change name');
}
$('.addName1').toggle();
$(/*TEXT FROM INPUT*/).appendTo(".p1");
});
polifitgsl.cpp：
#ifndef _POLIFITGSL_H
#define _POLIFITGSL_H
#include <gsl/gsl_multifit.h>
#include <stdbool.h>
#include <math.h>
bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store); /* n, p */
#endif
main.cpp（注意我用自己的x替换了x的样本编号）：
#include "polifitgsl.h"
bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store) /* n, p */
{
gsl_multifit_linear_workspace *ws;
gsl_matrix *cov, *X;
gsl_vector *y, *c;
double chisq;
int i, j;
X = gsl_matrix_alloc(obs, degree);
y = gsl_vector_alloc(obs);
c = gsl_vector_alloc(degree);
cov = gsl_matrix_alloc(degree, degree);
for(i=0; i < obs; i++) {
for(j=0; j < degree; j++) {
gsl_matrix_set(X, i, j, pow(dx[i], j));
}
gsl_vector_set(y, i, dy[i]);
}
ws = gsl_multifit_linear_alloc(obs, degree);
gsl_multifit_linear(X, y, c, cov, &chisq, ws);
/* store result ... */
for(i=0; i < degree; i++)
{
store[i] = gsl_vector_get(c, i);
}
gsl_multifit_linear_free(ws);
gsl_matrix_free(X);
gsl_matrix_free(cov);
gsl_vector_free(y);
gsl_vector_free(c);
return true; /* we do not "analyse" the result (cov matrix mainly)
to know if the fit is "good" */
}
这是输出：
#include <stdio.h>
#include "polifitgsl.h"
#define NP 11
double x[] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19};
double y[] = {98.02, 98.01, 98.01, 98.02, 97.98, 97.97, 97.96, 97.94, 97.96, 97.96, 97.97, 97.97, 97.94, 97.94, 97.94, 97.92, 97.96, 97.9, 97.85, 97.9};
#define DEGREE 3
double coeff[DEGREE];
int main()
{
int i;
polynomialfit(NP, DEGREE, x, y, coeff);
for(i=0; i < DEGREE; i++) {
printf("%lf\n", coeff[i]);
}
return 0;
}
这样就给了我系数。但我真正想要的是实际的拟合点。在JavaScript中，我使用regression package来计算点数：
98.030909
-0.016182
0.000909
产生：
var regression = require('regression');
var calculateRegression = function(values, degree) {
var data = [];
var regressionOutput;
var valuesCount = values.length;
var i = 0;
// Format the data in a way the regression library expects.
for (i = 0; i < valuesCount; i++) {
data[i] = [i, values[i]];
}
// Use the library to calculate the regression.
regressionOutput = regression('polynomial', data, degree);
return regressionOutput;
};
var y = [98.02, 98.01, 98.01, 98.02, 97.98, 97.97, 97.96, 97.94, 97.96, 97.96, 97.97, 97.97, 97.94, 97.94, 97.94, 97.92, 97.96, 97.9, 97.85, 97.9];
console.log(calculateRegression(y, 3));
（注意JavaScript中存在浮点问题，因此数字不完全准确。）
{ equation:
[ 98.02987916431594,
-0.017378390369880512,
0.0015748071645344357,
-0.00005721503635571101 ],
points:
[ [ 0, 98.02987916431594 ],
[ 1, 98.01401836607424 ],
[ 2, 98.00096389194348 ],
[ 3, 97.9903724517055 ],
[ 4, 97.98190075514219 ],
[ 5, 97.97520551203543 ],
[ 6, 97.96994343216707 ],
[ 7, 97.96577122531896 ],
[ 8, 97.96234560127297 ],
[ 9, 97.959323269811 ],
[ 10, 97.95636094071487 ],
[ 11, 97.95311532376647 ],
[ 12, 97.94924312874768 ],
[ 13, 97.94440106544033 ],
[ 14, 97.93824584362629 ],
[ 15, 97.93043417308745 ],
[ 16, 97.92062276360569 ],
[ 17, 97.90846832496283 ],
[ 18, 97.89362756694074 ],
[ 19, 97.87575719932133 ] ],
string: 'y = 0x^3 + 0x^2 + -0.02x + 98.03' }
这是我想要使用gsl生成的内容。有没有办法可以做到这一点？
答案 0 :(得分：1)
我遇到了同样的问题，并采取了上述答案并添加了一个直接计算的解决方案（没有gsl）取自http://www.cplusplus.com/forum/general/181580/
下面，您将找到一个独立的测试程序，其中包含基于gsl和直接计算的解决方案。
我已经完成了一些分析运行，直接计算的性能在我的系统上是令人印象深刻的高65倍，对于相应功能的1000次调用，5.22s与0.08s。
作为旁注，直接计算使用Cramer的规则，因此您应该注意病态条件数据。通常情况下我会避免使用Cramer，但是对于3x3系统使用全线性系统求解器对我来说似乎有点过分了。
/*
* =====================================================================================
*
* Filename: polyfit.cpp
*
* Description: Least squares fit of second order polynomials
* Test program using gsl and direct calculation
* Version: 1.0
* Created: 2017-07-17 09:32:55
* Compiler: gcc
*
* Author: Bernhard Brunner, brb_blog@epr.ch
* References: This code was merged, adapted and optimized from these two sources:
* http://www.cplusplus.com/forum/general/181580/
* https://stackoverflow.com/questions/36522882/how-can-i-use-gsl-to-calculate-polynomial-regression-data-points
* http://brb.epr.ch/blog/blog:least_squares_regression_of_parabola
* Build: compile and link using
* g++ -c -o polifitgsl.o polifitgsl.cpp
* gcc polifitgsl.o -lgsl -lm -lblas -o polifitgsl
*
* Profiling:
* valgrind --tool=callgrind ./polifitgsl
* kcachegrind
*
* polynomialfit takes 5.22s for 1000 calls
* findQuadCoefficients takes 0.08s for 1000 calls
* 65x faster
* =====================================================================================
*/
#include <stdio.h>
#include <gsl/gsl_multifit.h>
#include <stdbool.h>
#include <math.h>
bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store); /* n, p */
double x[] = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19};
double y[] = { 98.02, 98.01, 98.01, 98.02, 97.98,
97.97, 97.96, 97.94, 97.96, 97.96,
97.97, 97.97, 97.94, 97.94, 97.94,
97.92, 97.96, 97.9, 97.85, 97.9 };
#define NP (sizeof(x)/sizeof(double)) // 20
#define DEGREE 3
double coeff[DEGREE];
bool findQuadCoefficients(double timeArray[], double valueArray[], double *coef, double &critPoint, int PointsNum){
const double S00=PointsNum;//points number
double S40=0, S10=0, S20=0, S30=0, S01=0, S11=0, S21 = 0;
// const double MINvalue = valueArray[0];
// const double MINtime = timeArray[0];
for (int i=0; i<PointsNum; i++ ){
double value = valueArray[i]; // - MINvalue); //normalizing
// cout << "i=" << i << " index=" << index << " value=" << value << endl;
int index = timeArray[i]; // - MINtime;
int index2 = index * index;
int index3 = index2 * index;
int index4 = index3 * index;
S40+= index4;
S30+= index3;
S20+= index2;
S10+= index;
S01 += value;
S11 += value*index;
S21 += value*index2;
}
double S20squared = S20*S20;
//minors M_ij=M_ji
double M11 = S20*S00 - S10*S10;
double M21 = S30*S00 - S20*S10;
double M22 = S40*S00 - S20squared;
double M31 = S30*S10 - S20squared;
double M32 = S40*S10 - S20*S30;
// double M33 = S40*S20 - pow(S30,2);
double discriminant = S40*M11 - S30*M21 + S20*M31;
// printf("discriminant :%lf\n", discriminant);
if (abs(discriminant) < .00000000001) return false;
double Da = S21*M11
-S11*M21
+S01*M31;
coef[2] = Da/discriminant;
// cout << "discriminant=" << discriminant;
// cout << " Da=" << Da;
double Db = -S21*M21
+S11*M22
-S01*M32;
coef[1] = Db/discriminant;
// cout << " Db=" << Db << endl;
double Dc = S40*(S20*S01 - S10*S11)
- S30*(S30*S01 - S10*S21)
+ S20*(S30*S11 - S20*S21);
coef[0] = Dc/discriminant;
// printf("c=%lf\n", c);
critPoint = -Db/(2*Da); // + MINtime; //-b/(2*a)= -Db/discriminant / (2*(Da/discriminant)) = -Db/(2*Da);
return true;
}
bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store) /* n, p */
{
gsl_multifit_linear_workspace *ws;
gsl_matrix *cov, *X;
gsl_vector *y, *c;
double chisq;
int i, j;
X = gsl_matrix_alloc(obs, degree);
y = gsl_vector_alloc(obs);
c = gsl_vector_alloc(degree);
cov = gsl_matrix_alloc(degree, degree);
for(i=0; i < obs; i++) {
for(j=0; j < degree; j++) {
gsl_matrix_set(X, i, j, pow(dx[i], j));
}
gsl_vector_set(y, i, dy[i]);
}
ws = gsl_multifit_linear_alloc(obs, degree);
gsl_multifit_linear(X, y, c, cov, &chisq, ws);
/* store result ... */
for(i=0; i < degree; i++) {
store[i] = gsl_vector_get(c, i);
}
gsl_multifit_linear_free(ws);
gsl_matrix_free(X);
gsl_matrix_free(cov);
gsl_vector_free(y);
gsl_vector_free(c);
return true; /* we do not "analyse" the result (cov matrix mainly)
to know if the fit is "good" */
}
void testcoeff(double *coeff)
{
printf ("\n polynomial coefficients\n");
for (int i = 0; i < DEGREE; i++) {
printf (" coeff[%d] : %11.7lf\n", i, coeff[i]);
}
putchar ('\n');
printf (" computed values:\n\n x, yi, yip\n");
for (unsigned i = 0; i < NP; i++) {
printf ("%2u,%.7lf,%.7lf\n", i,
y[i],
i*i*coeff[2] + i*coeff[1] + coeff[0]);
}
putchar ('\n');
}
int main (void)
{
#define ITER 1000
for (int i=0; i< ITER; i++) {
polynomialfit (NP, DEGREE, x, y, coeff);
}
testcoeff(coeff);
double sx;
for (int i=0; i< ITER; i++) {
findQuadCoefficients(x, y, coeff, sx, NP);
}
printf("critical point %lf\n", sx);
testcoeff(coeff);
return 0;
}
参考文献：