#include <stdio.h>
#include <math.h>
const int TERMS = 7;
const float PI = 3.14159265358979;
int fact(int n) {
return n<= 0 ? 1 : n * fact(n-1);
}
double sine(int x) {
double rad = x * (PI / 180);
double sin = 0;
int n;
for(n = 0; n < TERMS; n++) { // That's Taylor series!!
sin += pow(-1, n) * pow(rad, (2 * n) + 1)/ fact((2 * n) + 1);
}
return sin;
}
double cosine(int x) {
double rad = x * (PI / 180);
double cos = 0;
int n;
for(n = 0; n < TERMS; n++) { // That's also Taylor series!
cos += pow(-1, n) * pow(rad, 2 * n) / fact(2 * n);
}
return cos;
}
int main(void){
int y;
scanf("%d",&y);
printf("sine(%d)= %lf\n",y, sine(y));
printf("cosine(%d)= %lf\n",y, cosine(y));
return 0;
}
The code above was implemented to compute sine and cosine using Taylor series.
I tried testing the code and it works fine for sine(120).
I am getting wrong answers for sine(240) and sine(300).
Can anyone help me find out why those errors occur?
You should calculate the functions in the first quadrant only [0, pi/2). Exploit the properties of the functions to get the values for other angles. For instance, for values of x between [pi/2, pi), sin(x) can be calculated by sin(pi - x).
The sine of 120 degrees, which is 40 past 90 degrees, is the same as 50 degrees: 40 degrees before 90. Sine starts at 0, then rises toward 1 at 90 degrees, and then falls again in a mirror image to zero at 180.
The negative sine values from pi to 2pi are just -sin(x - pi). I'd handle everything by this recursive definition:
sin(x):
cases x of:
[0, pi/2) -> calculate (Taylor or whatever)
[pi/2, pi) -> sin(pi - x)
[pi/2, 2pi) -> -sin(x - pi)
< 0 -> sin(-x)
>= 2pi -> sin(fmod(x, 2pi)) // floating-point remainder
A similar approach for cos, using identity cases appropriate for it.
The key point is:
TERMS is too small to have proper precision. And if you increase TERMS, you have to change fact implementation as it will likely overflow when working with int.
I would use a sign to toggle the -1 power instead of pow(-1,n) overkill.
Then use double for the value of PI to avoid losing too many decimals
Then for high values, you should increase the number of terms (this is the main issue). using long long for your factorial method or you get overflow. I set 10 and get proper results:
#include <stdio.h>
#include <math.h>
const int TERMS = 10;
const double PI = 3.14159265358979;
long long fact(int n) {
return n<= 0 ? 1 : n * fact(n-1);
}
double powd(double x,int n) {
return n<= 0 ? 1 : x * powd(x,n-1);
}
double sine(int x) {
double rad = x * (PI / 180);
double sin = 0;
int n;
int sign = 1;
for(n = 0; n < TERMS; n++) { // That's Taylor series!!
sin += sign * powd(rad, (2 * n) + 1)/ fact((2 * n) + 1);
sign = -sign;
}
return sin;
}
double cosine(int x) {
double rad = x * (PI / 180);
double cos = 0;
int n;
int sign = 1;
for(n = 0; n < TERMS; n++) { // That's also Taylor series!
cos += sign * powd(rad, 2 * n) / fact(2 * n);
sign = -sign;
}
return cos;
}
int main(void){
int y;
scanf("%d",&y);
printf("sine(%d)= %lf\n",y, sine(y));
printf("cosine(%d)= %lf\n",y, cosine(y));
return 0;
}
result:
240
sine(240)= -0.866026
cosine(240)= -0.500001
Notes:
my recusive implementation of pow using successive multiplications is probably not needed, since we're dealing with floating point. It introduces accumulation error if n is big.
fact could be using floating point to allow bigger numbers and better precision. Actually I suggested long long but it would be better not to assume that the size will be enough. Better use standard type like int64_t for that.
fact and pow results could be pre-computed/hardcoded as well. This would save computation time.
const double TERMS = 14;
const double PI = 3.14159265358979;
double fact(double n) {return n <= 0.0 ? 1 : n * fact(n - 1);}
double sine(double x)
{
double rad = x * (PI / 180);
rad = fmod(rad, 2 * PI);
double sin = 0;
for (double n = 0; n < TERMS; n++)
sin += pow(-1, n) * pow(rad, (2 * n) + 1) / fact((2 * n) + 1);
return sin;
}
double cosine(double x)
{
double rad = x * (PI / 180);
rad = fmod(rad,2*PI);
double cos = 0;
for (double n = 0; n < TERMS; n++)
cos += pow(-1, n) * pow(rad, 2 * n) / fact(2 * n);
return cos;
}
int main()
{
printf("sine(240)= %lf\n", sine(240));
printf("cosine(300)= %lf\n",cosine(300));
}
I am trying to write a simple ray tracer. The final image should like this: I have read stuff about it and below is what I am doing:
create an empty image (to fill each pixel, via ray tracing)
for each pixel [for each row, each column]
create the equation of the ray emanating from our pixel
trace() ray:
if ray intersects SPHERE
compute local shading (including shadow determination)
return color;
Now, the scene data is like: It sets a gray sphere of radius 1 at (0,0,-3). It sets a white light source at the origin.
2
amb: 0.3 0.3 0.3
sphere
pos: 0.0 0.0 -3.0
rad: 1
dif: 0.3 0.3 0.3
spe: 0.5 0.5 0.5
shi: 1
light
pos: 0 0 0
col: 1 1 1
Mine looks very weird :
//check ray intersection with the sphere
boolean intersectsWithSphere(struct point rayPosition, struct point rayDirection, Sphere sp,float* t){
//float a = (rayDirection.x * rayDirection.x) + (rayDirection.y * rayDirection.y) +(rayDirection.z * rayDirection.z);
// value for a is 1 since rayDirection vector is normalized
double radius = sp.radius;
double xc = sp.position[0];
double yc =sp.position[1];
double zc =sp.position[2];
double xo = rayPosition.x;
double yo = rayPosition.y;
double zo = rayPosition.z;
double xd = rayDirection.x;
double yd = rayDirection.y;
double zd = rayDirection.z;
double b = 2 * ((xd*(xo-xc))+(yd*(yo-yc))+(zd*(zo-zc)));
double c = (xo-xc)*(xo-xc) + (yo-yc)*(yo-yc) + (zo-zc)*(zo-zc) - (radius * radius);
float D = b*b + (-4.0f)*c;
//ray does not intersect the sphere
if(D < 0 ){
return false;
}
D = sqrt(D);
float t0 = (-b - D)/2 ;
float t1 = (-b + D)/2;
//printf("D=%f",D);
//printf(" t0=%f",t0);
//printf(" t1=%f\n",t1);
if((t0 > 0) && (t1 > 0)){
*t = min(t0,t1);
return true;
}
else {
*t = 0;
return false;
}
}
Below is the trace() function:
unsigned char* trace(struct point rayPosition, struct point rayDirection, Sphere * totalspheres) {
struct point tempRayPosition = rayPosition;
struct point tempRayDirection = rayDirection;
float f=0;
float tnear = INFINITY;
boolean sphereIntersectionFound = false;
int sphereIndex = -1;
for(int i=0; i < num_spheres ; i++){
float t = INFINITY;
if(intersectsWithSphere(tempRayPosition,tempRayDirection,totalspheres[i],&t)){
if(t < tnear){
tnear = t;
sphereIntersectionFound = true;
sphereIndex = i;
}
}
}
if(sphereIndex < 0){
//printf("No interesection found\n");
mycolor[0] = 1;
mycolor[1] = 1;
mycolor[2] = 1;
return mycolor;
}
else {
Sphere sp = totalspheres[sphereIndex];
//intersection point
hitPoint[0].x = tempRayPosition.x + tempRayDirection.x * tnear;
hitPoint[0].y = tempRayPosition.y + tempRayDirection.y * tnear;
hitPoint[0].z = tempRayPosition.z + tempRayDirection.z * tnear;
//normal at the intersection point
normalAtHitPoint[0].x = (hitPoint[0].x - totalspheres[sphereIndex].position[0])/ totalspheres[sphereIndex].radius;
normalAtHitPoint[0].y = (hitPoint[0].y - totalspheres[sphereIndex].position[1])/ totalspheres[sphereIndex].radius;
normalAtHitPoint[0].z = (hitPoint[0].z - totalspheres[sphereIndex].position[2])/ totalspheres[sphereIndex].radius;
normalizedNormalAtHitPoint[0] = normalize(normalAtHitPoint[0]);
for(int j=0; j < num_lights ; j++) {
for(int k=0; k < num_spheres ; k++){
shadowRay[0].x = lights[j].position[0] - hitPoint[0].x;
shadowRay[0].y = lights[j].position[1] - hitPoint[0].y;
shadowRay[0].z = lights[j].position[2] - hitPoint[0].z;
normalizedShadowRay[0] = normalize(shadowRay[0]);
//R = 2 * ( N dot L) * N - L
reflectionRay[0].x = - 2 * dot(normalizedShadowRay[0],normalizedNormalAtHitPoint[0]) * normalizedNormalAtHitPoint[0].x +normalizedShadowRay[0].x;
reflectionRay[0].y = - 2 * dot(normalizedShadowRay[0],normalizedNormalAtHitPoint[0]) * normalizedNormalAtHitPoint[0].y +normalizedShadowRay[0].y;
reflectionRay[0].z = - 2 * dot(normalizedShadowRay[0],normalizedNormalAtHitPoint[0]) * normalizedNormalAtHitPoint[0].z +normalizedShadowRay[0].z;
normalizeReflectionRay[0] = normalize(reflectionRay[0]);
struct point temp;
temp.x = hitPoint[0].x + (shadowRay[0].x * 0.0001 );
temp.y = hitPoint[0].y + (shadowRay[0].y * 0.0001);
temp.z = hitPoint[0].z + (shadowRay[0].z * 0.0001);
struct point ntemp = normalize(temp);
float f=0;
struct point tempHitPoint;
tempHitPoint.x = hitPoint[0].x + 0.001;
tempHitPoint.y = hitPoint[0].y + 0.001;
tempHitPoint.z = hitPoint[0].z + 0.001;
if(intersectsWithSphere(hitPoint[0],ntemp,totalspheres[k],&f)){
// if(intersectsWithSphere(tempHitPoint,ntemp,totalspheres[k],&f)){
printf("In shadow\n");
float r = lights[j].color[0];
float g = lights[j].color[1];
float b = lights[j].color[2];
mycolor[0] = ambient_light[0] + r;
mycolor[1] = ambient_light[1] + g;
mycolor[2] = ambient_light[2] + b;
return mycolor;
} else {
// point is not is shadow , use Phong shading to determine the color of the point.
//I = lightColor * (kd * (L dot N) + ks * (R dot V) ^ sh)
//(for each color channel separately; note that if L dot N < 0, you should clamp L dot N to zero; same for R dot V)
float x = dot(normalizedShadowRay[0],normalizedNormalAtHitPoint[0]);
if(x < 0)
x = 0;
V[0].x = - rayDirection.x;
V[0].x = - rayDirection.y;
V[0].x = - rayDirection.z;
normalizedV[0] = normalize(V[0]);
float y = dot(normalizeReflectionRay[0],normalizedV[0]);
if(y < 0)
y = 0;
float ar = totalspheres[sphereIndex].color_diffuse[0] * x;
float br = totalspheres[sphereIndex].color_specular[0] * pow(y,totalspheres[sphereIndex].shininess);
float r = lights[j].color[0] * (ar+br);
//----------------------------------------------------------------------------------
float bg = totalspheres[sphereIndex].color_specular[1] * pow(y,totalspheres[sphereIndex].shininess);
float ag = totalspheres[sphereIndex].color_diffuse[1] * x;
float g = lights[j].color[1] * (ag+bg);
//----------------------------------------------------------------------------------
float bb = totalspheres[sphereIndex].color_specular[2] * pow(y,totalspheres[sphereIndex].shininess);
float ab = totalspheres[sphereIndex].color_diffuse[2] * x;
float b = lights[j].color[2] * (ab+bb);
mycolor[0] = r + ambient_light[0];
mycolor[1] = g + ambient_light[1];
mycolor[2] = b+ ambient_light[2];
return mycolor;
}
}
}
}
}
The code calling trace() looks like :
void draw_scene()
{
//Aspect Ratio
double a = WIDTH / HEIGHT;
double angel = tan(M_PI * 0.5 * fov/ 180);
ray[0].x = 0.0;
ray[0].y = 0.0;
ray[0].z = 0.0;
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
unsigned int x,y;
float sx, sy;
for(x=0;x < WIDTH;x++)
{
glPointSize(2.0);
glBegin(GL_POINTS);
for(y=0;y < HEIGHT;y++)
{
sx = (((x + 0.5) / WIDTH) * 2.0 ) - 1;
sy = (((y + 0.5) / HEIGHT) * 2.0 ) - 1;;
sx = sx * angel * a;
sy = sy * angel;
//set ray direction
ray[1].x = sx;
ray[1].y = sy;
ray[1].z = -1;
normalizedRayDirection[0] = normalize(ray[1]);
unsigned char* color = trace(ray[0],normalizedRayDirection[0],spheres);
unsigned char x1 = color[0] * 255;
unsigned char y1 = color[1] * 255;
unsigned char z1 = color[2] * 255;
plot_pixel(x,y,x1 %256,y1%256,z1%256);
}
glEnd();
glFlush();
}
}
There could be many, many problems with the code/understanding.
I haven't taken the time to understand all your code, and I'm definitely not a graphics expert, but I believe the problem you have is called "surface acne". In this case it's probably happening because your shadow rays are intersecting with the object itself. What I did in my code to fix this is add epsilon * hitPoint.normal to the shadow ray origin. This effectively moves the ray away from your object a bit, so they don't intersect.
The value I'm using for epsilon is the square root of 1.19209290 * 10^-7, as that is the square root of a constant called EPSILON that is defined in the particular language I'm using.
What possible reason do you have for doing this (in the non-shadow branch of trace (...)):
V[0].x = - rayDirection.x;
V[0].x = - rayDirection.y;
V[0].x = - rayDirection.z;
You might as well comment out the first two computations since you write the results of each to the same component. I think you probably meant to do this instead:
V[0].x = - rayDirection.x;
V[0].y = - rayDirection.y;
V[0].z = - rayDirection.z;
That said, you should also avoid using GL_POINT primitives to cover a 2x2 pixel quad. Point primitives are not guaranteed to be square, and OpenGL implementations are not required to support any size other than 1.0. In practice, most support 1.0 - ~64.0 but glDrawPixels (...) is a much better way of writing 2x2 pixels, since it skips primitive assembly and the above mentioned limitations. You are using immediate mode in this example anyway, so glRasterPos (...) and glDrawPixels (...) are still a valid approach.
It seems you are implementing the formula here, but you deviate at the end from the direction the article takes.
First the article warns that D & b can be very close in value, so that -b + D gets you a very limited number. They suggest an alternative.
Also, you are testing that both t0 & t1 > 0. This doesn't have to be true for you to hit the sphere, you could be inside of it (though you obviously should not be in your test scene).
Finally, I would add a test at the beginning to confirm that the direction vector is normalized. I've messed that up more than once in my renderers.
I am trying to generate an array of n points that are equidistant from each other and lie on a circle in C. Basically, I need to be able to pass a function the number of points that I would like to generate and get back an array of points.
It's been a really long time since I've done C/C++, so I've had a stab at this more to see how I got on with it, but here's some code that will calculate the points for you. (It's a VS2010 console application)
// CirclePoints.cpp : Defines the entry point for the console application.
//
#include "stdafx.h"
#include "stdio.h"
#include "math.h"
int _tmain()
{
int points = 8;
double radius = 100;
double step = ((3.14159265 * 2) / points);
double x, y, current = 0;
for (int i = 0; i < points; i++)
{
x = sin(current) * radius;
y = cos(current) * radius;
printf("point: %d x:%lf y:%lf\n", i, x, y);
current += step;
}
return 0;
}
Try something like this:
void make_circle(float *output, size_t num, float radius)
{
size_t i;
for(i = 0; i < num; i++)
{
const float angle = 2 * M_PI * i / num;
*output++ = radius * cos(angle);
*output++ = radius * sin(angle);
}
}
This is untested, there might be an off-by-one hiding in the angle step calculation but it should be close.
This assumes I understood the question correctly, of course.
UPDATE: Redid the angle computation to not be incrementing, to reduce float precision loss due to repeated addition.
Here's a solution, somewhat optimized, untested. Error can accumulate, but using double rather than float probably more than makes up for it except with extremely large values of n.
void make_circle(double *dest, size_t n, double r)
{
double x0 = cos(2*M_PI/n), y0 = sin(2*M_PI/n), x=x0, y=y0, tmp;
for (;;) {
*dest++ = r*x;
*dest++ = r*y;
if (!--n) break;
tmp = x*x0 - y*y0;
y = x*y0 + y*x0;
x = tmp;
}
}
You have to solve this in c language:
In an x-y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that
(x - a)^2 + (y - b)^2 = r^2
Here's a javascript implementation that also takes an optional center point.
function circlePoints (radius, numPoints, centerX, centerY) {
centerX = centerX || 0;
centerY = centerY || 0;
var
step = (Math.PI * 2) / numPoints,
current = 0,
i = 0,
results = [],
x, y;
for (; i < numPoints; i += 1) {
x = centerX + Math.sin(current) * radius;
y = centerY + Math.cos(current) * radius;
results.push([x,y]);
console.log('point %d # x:%d, y:%d', i, x, y);
current += step;
}
return results;
}
I am trying to implement a linear least squares fit onto 2 arrays of data: time vs amplitude. The only technique I know so far is to test all of the possible m and b points in (y = m*x+b) and then find out which combination fits my data best so that it has the least error. However, I think iterating so many combinations is sometimes useless because it tests out everything. Are there any techniques to speed up the process that I don't know about? Thanks.
Try this code. It fits y = mx + b to your (x,y) data.
The arguments to linreg are
linreg(int n, REAL x[], REAL y[], REAL* b, REAL* m, REAL* r)
n = number of data points
x,y = arrays of data
*b = output intercept
*m = output slope
*r = output correlation coefficient (can be NULL if you don't want it)
The return value is 0 on success, !=0 on failure.
Here's the code
#include "linreg.h"
#include <stdlib.h>
#include <math.h> /* math functions */
//#define REAL float
#define REAL double
inline static REAL sqr(REAL x) {
return x*x;
}
int linreg(int n, const REAL x[], const REAL y[], REAL* m, REAL* b, REAL* r){
REAL sumx = 0.0; /* sum of x */
REAL sumx2 = 0.0; /* sum of x**2 */
REAL sumxy = 0.0; /* sum of x * y */
REAL sumy = 0.0; /* sum of y */
REAL sumy2 = 0.0; /* sum of y**2 */
for (int i=0;i<n;i++){
sumx += x[i];
sumx2 += sqr(x[i]);
sumxy += x[i] * y[i];
sumy += y[i];
sumy2 += sqr(y[i]);
}
REAL denom = (n * sumx2 - sqr(sumx));
if (denom == 0) {
// singular matrix. can't solve the problem.
*m = 0;
*b = 0;
if (r) *r = 0;
return 1;
}
*m = (n * sumxy - sumx * sumy) / denom;
*b = (sumy * sumx2 - sumx * sumxy) / denom;
if (r!=NULL) {
*r = (sumxy - sumx * sumy / n) / /* compute correlation coeff */
sqrt((sumx2 - sqr(sumx)/n) *
(sumy2 - sqr(sumy)/n));
}
return 0;
}
Example
You can run this example online.
int main()
{
int n = 6;
REAL x[6]= {1, 2, 4, 5, 10, 20};
REAL y[6]= {4, 6, 12, 15, 34, 68};
REAL m,b,r;
linreg(n,x,y,&m,&b,&r);
printf("m=%g b=%g r=%g\n",m,b,r);
return 0;
}
Here is the output
m=3.43651 b=-0.888889 r=0.999192
Here is the Excel plot and linear fit (for verification).
All values agree exactly with the C code above (note C code returns r while Excel returns R**2).
There are efficient algorithms for least-squares fitting; see Wikipedia for details. There are also libraries that implement the algorithms for you, likely more efficiently than a naive implementation would do; the GNU Scientific Library is one example, but there are others under more lenient licenses as well.
From Numerical Recipes: The Art of Scientific Computing in (15.2) Fitting Data to a Straight Line:
Linear Regression:
Consider the problem of fitting a set of N data points (xi, yi) to a straight-line model:
Assume that the uncertainty: sigmai associated with each yi and that the xi’s (values of the dependent variable) are known exactly. To measure how well the model agrees with the data, we use the chi-square function, which in this case is:
The above equation is minimized to determine a and b. This is done by finding the derivative of the above equation with respect to a and b, equate them to zero and solve for a and b. Then we estimate the probable uncertainties in the estimates of a and b, since obviously the measurement errors in the data must introduce some uncertainty in the determination of those parameters. Additionally, we must estimate the goodness-of-fit of the data to the
model. Absent this estimate, we have not the slightest indication that the parameters a and b in the model have any meaning at all.
The below struct performs the mentioned calculations:
struct Fitab {
// Object for fitting a straight line y = a + b*x to a set of
// points (xi, yi), with or without available
// errors sigma i . Call one of the two constructors to calculate the fit.
// The answers are then available as the variables:
// a, b, siga, sigb, chi2, and either q or sigdat.
int ndata;
double a, b, siga, sigb, chi2, q, sigdat; // Answers.
vector<double> &x, &y, &sig;
// Constructor.
Fitab(vector<double> &xx, vector<double> &yy, vector<double> &ssig)
: ndata(xx.size()), x(xx), y(yy), sig(ssig), chi2(0.), q(1.), sigdat(0.)
{
// Given a set of data points x[0..ndata-1], y[0..ndata-1]
// with individual standard deviations sig[0..ndata-1],
// sets a,b and their respective probable uncertainties
// siga and sigb, the chi-square: chi2, and the goodness-of-fit
// probability: q
Gamma gam;
int i;
double ss=0., sx=0., sy=0., st2=0., t, wt, sxoss; b=0.0;
for (i=0;i < ndata; i++) { // Accumulate sums ...
wt = 1.0 / SQR(sig[i]); //...with weights
ss += wt;
sx += x[i]*wt;
sy += y[i]*wt;
}
sxoss = sx/ss;
for (i=0; i < ndata; i++) {
t = (x[i]-sxoss) / sig[i];
st2 += t*t;
b += t*y[i]/sig[i];
}
b /= st2; // Solve for a, b, sigma-a, and simga-b.
a = (sy-sx*b) / ss;
siga = sqrt((1.0+sx*sx/(ss*st2))/ss);
sigb = sqrt(1.0/st2); // Calculate chi2.
for (i=0;i<ndata;i++) chi2 += SQR((y[i]-a-b*x[i])/sig[i]);
if (ndata>2) q=gam.gammq(0.5*(ndata-2),0.5*chi2); // goodness of fit
}
// Constructor.
Fitab(vector<double> &xx, vector<double> &yy)
: ndata(xx.size()), x(xx), y(yy), sig(xx), chi2(0.), q(1.), sigdat(0.)
{
// As above, but without known errors (sig is not used).
// The uncertainties siga and sigb are estimated by assuming
// equal errors for all points, and that a straight line is
// a good fit. q is returned as 1.0, the normalization of chi2
// is to unit standard deviation on all points, and sigdat
// is set to the estimated error of each point.
int i;
double ss,sx=0.,sy=0.,st2=0.,t,sxoss;
b=0.0; // Accumulate sums ...
for (i=0; i < ndata; i++) {
sx += x[i]; // ...without weights.
sy += y[i];
}
ss = ndata;
sxoss = sx/ss;
for (i=0;i < ndata; i++) {
t = x[i]-sxoss;
st2 += t*t;
b += t*y[i];
}
b /= st2; // Solve for a, b, sigma-a, and sigma-b.
a = (sy-sx*b)/ss;
siga=sqrt((1.0+sx*sx/(ss*st2))/ss);
sigb=sqrt(1.0/st2); // Calculate chi2.
for (i=0;i<ndata;i++) chi2 += SQR(y[i]-a-b*x[i]);
if (ndata > 2) sigdat=sqrt(chi2/(ndata-2));
// For unweighted data evaluate typical
// sig using chi2, and adjust
// the standard deviations.
siga *= sigdat;
sigb *= sigdat;
}
};
where struct Gamma:
struct Gamma : Gauleg18 {
// Object for incomplete gamma function.
// Gauleg18 provides coefficients for Gauss-Legendre quadrature.
static const Int ASWITCH=100; When to switch to quadrature method.
static const double EPS; // See end of struct for initializations.
static const double FPMIN;
double gln;
double gammp(const double a, const double x) {
// Returns the incomplete gamma function P(a,x)
if (x < 0.0 || a <= 0.0) throw("bad args in gammp");
if (x == 0.0) return 0.0;
else if ((Int)a >= ASWITCH) return gammpapprox(a,x,1); // Quadrature.
else if (x < a+1.0) return gser(a,x); // Use the series representation.
else return 1.0-gcf(a,x); // Use the continued fraction representation.
}
double gammq(const double a, const double x) {
// Returns the incomplete gamma function Q(a,x) = 1 - P(a,x)
if (x < 0.0 || a <= 0.0) throw("bad args in gammq");
if (x == 0.0) return 1.0;
else if ((Int)a >= ASWITCH) return gammpapprox(a,x,0); // Quadrature.
else if (x < a+1.0) return 1.0-gser(a,x); // Use the series representation.
else return gcf(a,x); // Use the continued fraction representation.
}
double gser(const Doub a, const Doub x) {
// Returns the incomplete gamma function P(a,x) evaluated by its series representation.
// Also sets ln (gamma) as gln. User should not call directly.
double sum,del,ap;
gln=gammln(a);
ap=a;
del=sum=1.0/a;
for (;;) {
++ap;
del *= x/ap;
sum += del;
if (fabs(del) < fabs(sum)*EPS) {
return sum*exp(-x+a*log(x)-gln);
}
}
}
double gcf(const Doub a, const Doub x) {
// Returns the incomplete gamma function Q(a, x) evaluated
// by its continued fraction representation.
// Also sets ln (gamma) as gln. User should not call directly.
int i;
double an,b,c,d,del,h;
gln=gammln(a);
b=x+1.0-a; // Set up for evaluating continued fraction
// by modified Lentz’s method with with b0 = 0.
c=1.0/FPMIN;
d=1.0/b;
h=d;
for (i=1;;i++) {
// Iterate to convergence.
an = -i*(i-a);
b += 2.0;
d=an*d+b;
if (fabs(d) < FPMIN) d=FPMIN;
c=b+an/c;
if (fabs(c) < FPMIN) c=FPMIN;
d=1.0/d;
del=d*c;
h *= del;
if (fabs(del-1.0) <= EPS) break;
}
return exp(-x+a*log(x)-gln)*h; Put factors in front.
}
double gammpapprox(double a, double x, int psig) {
// Incomplete gamma by quadrature. Returns P(a,x) or Q(a, x),
// when psig is 1 or 0, respectively. User should not call directly.
int j;
double xu,t,sum,ans;
double a1 = a-1.0, lna1 = log(a1), sqrta1 = sqrt(a1);
gln = gammln(a);
// Set how far to integrate into the tail:
if (x > a1) xu = MAX(a1 + 11.5*sqrta1, x + 6.0*sqrta1);
else xu = MAX(0.,MIN(a1 - 7.5*sqrta1, x - 5.0*sqrta1));
sum = 0;
for (j=0;j<ngau;j++) { // Gauss-Legendre.
t = x + (xu-x)*y[j];
sum += w[j]*exp(-(t-a1)+a1*(log(t)-lna1));
}
ans = sum*(xu-x)*exp(a1*(lna1-1.)-gln);
return (psig?(ans>0.0? 1.0-ans:-ans):(ans>=0.0? ans:1.0+ans));
}
double invgammp(Doub p, Doub a);
// Inverse function on x of P(a,x) .
};
const Doub Gamma::EPS = numeric_limits<Doub>::epsilon();
const Doub Gamma::FPMIN = numeric_limits<Doub>::min()/EPS
and stuct Gauleg18:
struct Gauleg18 {
// Abscissas and weights for Gauss-Legendre quadrature.
static const Int ngau = 18;
static const Doub y[18];
static const Doub w[18];
};
const Doub Gauleg18::y[18] = {0.0021695375159141994,
0.011413521097787704,0.027972308950302116,0.051727015600492421,
0.082502225484340941, 0.12007019910960293,0.16415283300752470,
0.21442376986779355, 0.27051082840644336, 0.33199876341447887,
0.39843234186401943, 0.46931971407375483, 0.54413605556657973,
0.62232745288031077, 0.70331500465597174, 0.78649910768313447,
0.87126389619061517, 0.95698180152629142};
const Doub Gauleg18::w[18] = {0.0055657196642445571,
0.012915947284065419,0.020181515297735382,0.027298621498568734,
0.034213810770299537,0.040875750923643261,0.047235083490265582,
0.053244713977759692,0.058860144245324798,0.064039797355015485
0.068745323835736408,0.072941885005653087,0.076598410645870640,
0.079687828912071670,0.082187266704339706,0.084078218979661945,
0.085346685739338721,0.085983275670394821};
and, finally fuinction Gamma::invgamp():
double Gamma::invgammp(double p, double a) {
// Returns x such that P(a,x) = p for an argument p between 0 and 1.
int j;
double x,err,t,u,pp,lna1,afac,a1=a-1;
const double EPS=1.e-8; // Accuracy is the square of EPS.
gln=gammln(a);
if (a <= 0.) throw("a must be pos in invgammap");
if (p >= 1.) return MAX(100.,a + 100.*sqrt(a));
if (p <= 0.) return 0.0;
if (a > 1.) {
lna1=log(a1);
afac = exp(a1*(lna1-1.)-gln);
pp = (p < 0.5)? p : 1. - p;
t = sqrt(-2.*log(pp));
x = (2.30753+t*0.27061)/(1.+t*(0.99229+t*0.04481)) - t;
if (p < 0.5) x = -x;
x = MAX(1.e-3,a*pow(1.-1./(9.*a)-x/(3.*sqrt(a)),3));
} else {
t = 1.0 - a*(0.253+a*0.12); and (6.2.9).
if (p < t) x = pow(p/t,1./a);
else x = 1.-log(1.-(p-t)/(1.-t));
}
for (j=0;j<12;j++) {
if (x <= 0.0) return 0.0; // x too small to compute accurately.
err = gammp(a,x) - p;
if (a > 1.) t = afac*exp(-(x-a1)+a1*(log(x)-lna1));
else t = exp(-x+a1*log(x)-gln);
u = err/t;
// Halley’s method.
x -= (t = u/(1.-0.5*MIN(1.,u*((a-1.)/x - 1))));
// Halve old value if x tries to go negative.
if (x <= 0.) x = 0.5*(x + t);
if (fabs(t) < EPS*x ) break;
}
return x;
}
Here is my version of a C/C++ function that does simple linear regression. The calculations follow the wikipedia article on simple linear regression. This is published as a single-header public-domain (MIT) library on github: simple_linear_regression. The library (.h file) is tested to work on Linux and Windows, and from C and C++ using -Wall -Werror and all -std versions supported by clang/gcc.
#define SIMPLE_LINEAR_REGRESSION_ERROR_INPUT_VALUE -2
#define SIMPLE_LINEAR_REGRESSION_ERROR_NUMERIC -3
int simple_linear_regression(const double * x, const double * y, const int n, double * slope_out, double * intercept_out, double * r2_out) {
double sum_x = 0.0;
double sum_xx = 0.0;
double sum_xy = 0.0;
double sum_y = 0.0;
double sum_yy = 0.0;
double n_real = (double)(n);
int i = 0;
double slope = 0.0;
double denominator = 0.0;
if (x == NULL || y == NULL || n < 2) {
return SIMPLE_LINEAR_REGRESSION_ERROR_INPUT_VALUE;
}
for (i = 0; i < n; ++i) {
sum_x += x[i];
sum_xx += x[i] * x[i];
sum_xy += x[i] * y[i];
sum_y += y[i];
sum_yy += y[i] * y[i];
}
denominator = n_real * sum_xx - sum_x * sum_x;
if (denominator == 0.0) {
return SIMPLE_LINEAR_REGRESSION_ERROR_NUMERIC;
}
slope = (n_real * sum_xy - sum_x * sum_y) / denominator;
if (slope_out != NULL) {
*slope_out = slope;
}
if (intercept_out != NULL) {
*intercept_out = (sum_y - slope * sum_x) / n_real;
}
if (r2_out != NULL) {
denominator = ((n_real * sum_xx) - (sum_x * sum_x)) * ((n_real * sum_yy) - (sum_y * sum_y));
if (denominator == 0.0) {
return SIMPLE_LINEAR_REGRESSION_ERROR_NUMERIC;
}
*r2_out = ((n_real * sum_xy) - (sum_x * sum_y)) * ((n_real * sum_xy) - (sum_x * sum_y)) / denominator;
}
return 0;
}
Usage example:
#define SIMPLE_LINEAR_REGRESSION_IMPLEMENTATION
#include "simple_linear_regression.h"
#include <stdio.h>
/* Some data that we want to find the slope, intercept and r2 for */
static const double x[] = { 1.47, 1.50, 1.52, 1.55, 1.57, 1.60, 1.63, 1.65, 1.68, 1.70, 1.73, 1.75, 1.78, 1.80, 1.83 };
static const double y[] = { 52.21, 53.12, 54.48, 55.84, 57.20, 58.57, 59.93, 61.29, 63.11, 64.47, 66.28, 68.10, 69.92, 72.19, 74.46 };
int main() {
double slope = 0.0;
double intercept = 0.0;
double r2 = 0.0;
int res = 0;
res = simple_linear_regression(x, y, sizeof(x) / sizeof(x[0]), &slope, &intercept, &r2);
if (res < 0) {
printf("Error: %s\n", simple_linear_regression_error_string(res));
return res;
}
printf("slope: %f\n", slope);
printf("intercept: %f\n", intercept);
printf("r2: %f\n", r2);
return 0;
}
The original example above worked well for me with slope and offset but I had a hard time with the corr coef. Maybe I don't have my parenthesis working the same as the assumed precedence? Anyway, with some help of other web pages I finally got values that match the linear trend-line in Excel. Thought I would share my code using Mark Lakata's variable names. Hope this helps.
double slope = ((n * sumxy) - (sumx * sumy )) / denom;
double intercept = ((sumy * sumx2) - (sumx * sumxy)) / denom;
double term1 = ((n * sumxy) - (sumx * sumy));
double term2 = ((n * sumx2) - (sumx * sumx));
double term3 = ((n * sumy2) - (sumy * sumy));
double term23 = (term2 * term3);
double r2 = 1.0;
if (fabs(term23) > MIN_DOUBLE) // Define MIN_DOUBLE somewhere as 1e-9 or similar
r2 = (term1 * term1) / term23;
as an assignment I had to code in C a simple linear regression using RMSE loss function. The program is dynamic and you can enter your own values and choose your own loss function which is for now limited to Root Mean Square Error. But first here are the algorithms I used:
now the code... you need gnuplot to display the chart, sudo apt install gnuplot
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <sys/types.h>
#define BUFFSIZE 64
#define MAXSIZE 100
static double vector_x[MAXSIZE] = {0};
static double vector_y[MAXSIZE] = {0};
static double vector_predict[MAXSIZE] = {0};
static double max_x;
static double max_y;
static double mean_x;
static double mean_y;
static double teta_0_intercept;
static double teta_1_grad;
static double RMSE;
static double r_square;
static double prediction;
static char intercept[BUFFSIZE];
static char grad[BUFFSIZE];
static char xrange[BUFFSIZE];
static char yrange[BUFFSIZE];
static char lossname_RMSE[BUFFSIZE] = "Simple Linear Regression using RMSE'";
static char cmd_gnu_0[BUFFSIZE] = "set title '";
static char cmd_gnu_1[BUFFSIZE] = "intercept = ";
static char cmd_gnu_2[BUFFSIZE] = "grad = ";
static char cmd_gnu_3[BUFFSIZE] = "set xrange [0:";
static char cmd_gnu_4[BUFFSIZE] = "set yrange [0:";
static char cmd_gnu_5[BUFFSIZE] = "f(x) = (grad * x) + intercept";
static char cmd_gnu_6[BUFFSIZE] = "plot f(x), 'data.temp' with points pointtype 7";
static char const *commands_gnuplot[] = {
cmd_gnu_0,
cmd_gnu_1,
cmd_gnu_2,
cmd_gnu_3,
cmd_gnu_4,
cmd_gnu_5,
cmd_gnu_6,
};
static size_t size;
static void user_input()
{
printf("Enter x,y vector size, MAX = 100\n");
scanf("%lu", &size);
if (size > MAXSIZE) {
printf("Wrong input size is too big\n");
user_input();
}
printf("vector's size is %lu\n", size);
size_t i;
for (i = 0; i < size; i++) {
printf("Enter vector_x[%ld] values\n", i);
scanf("%lf", &vector_x[i]);
}
for (i = 0; i < size; i++) {
printf("Enter vector_y[%ld] values\n", i);
scanf("%lf", &vector_y[i]);
}
}
static void display_vector()
{
size_t i;
for (i = 0; i < size; i++){
printf("vector_x[%lu] = %lf\t", i, vector_x[i]);
printf("vector_y[%lu] = %lf\n", i, vector_y[i]);
}
}
static void concatenate(char p[], char q[]) {
int c;
int d;
c = 0;
while (p[c] != '\0') {
c++;
}
d = 0;
while (q[d] != '\0') {
p[c] = q[d];
d++;
c++;
}
p[c] = '\0';
}
static void compute_mean_x_y()
{
size_t i;
double tmp_x = 0.0;
double tmp_y = 0.0;
for (i = 0; i < size; i++) {
tmp_x += vector_x[i];
tmp_y += vector_y[i];
}
mean_x = tmp_x / size;
mean_y = tmp_y / size;
printf("mean_x = %lf\n", mean_x);
printf("mean_y = %lf\n", mean_y);
}
static void compute_teta_1_grad()
{
double numerator = 0.0;
double denominator = 0.0;
double tmp1 = 0.0;
double tmp2 = 0.0;
size_t i;
for (i = 0; i < size; i++) {
numerator += (vector_x[i] - mean_x) * (vector_y[i] - mean_y);
}
for (i = 0; i < size; i++) {
tmp1 = vector_x[i] - mean_x;
tmp2 = tmp1 * tmp1;
denominator += tmp2;
}
teta_1_grad = numerator / denominator;
printf("teta_1_grad = %lf\n", teta_1_grad);
}
static void compute_teta_0_intercept()
{
teta_0_intercept = mean_y - (teta_1_grad * mean_x);
printf("teta_0_intercept = %lf\n", teta_0_intercept);
}
static void compute_prediction()
{
size_t i;
for (i = 0; i < size; i++) {
vector_predict[i] = teta_0_intercept + (teta_1_grad * vector_x[i]);
printf("y^[%ld] = %lf\n", i, vector_predict[i]);
}
printf("\n");
}
static void compute_RMSE()
{
compute_prediction();
double error = 0;
size_t i;
for (i = 0; i < size; i++) {
error = (vector_predict[i] - vector_y[i]) * (vector_predict[i] - vector_y[i]);
printf("error y^[%ld] = %lf\n", i, error);
RMSE += error;
}
/* mean */
RMSE = RMSE / size;
/* square root mean */
RMSE = sqrt(RMSE);
printf("\nRMSE = %lf\n", RMSE);
}
static void compute_loss_function()
{
int input = 0;
printf("Which loss function do you want to use?\n");
printf(" 1 - RMSE\n");
scanf("%d", &input);
switch(input) {
case 1:
concatenate(cmd_gnu_0, lossname_RMSE);
compute_RMSE();
printf("\n");
break;
default:
printf("Wrong input try again\n");
compute_loss_function(size);
}
}
static void compute_r_square(size_t size)
{
double num_err = 0.0;
double den_err = 0.0;
size_t i;
for (i = 0; i < size; i++) {
num_err += (vector_y[i] - vector_predict[i]) * (vector_y[i] - vector_predict[i]);
den_err += (vector_y[i] - mean_y) * (vector_y[i] - mean_y);
}
r_square = 1 - (num_err/den_err);
printf("R_square = %lf\n", r_square);
}
static void compute_predict_for_x()
{
double x = 0.0;
printf("Please enter x value\n");
scanf("%lf", &x);
prediction = teta_0_intercept + (teta_1_grad * x);
printf("y^ if x = %lf -> %lf\n",x, prediction);
}
static void compute_max_x_y()
{
size_t i;
double tmp1= 0.0;
double tmp2= 0.0;
for (i = 0; i < size; i++) {
if (vector_x[i] > tmp1) {
tmp1 = vector_x[i];
max_x = vector_x[i];
}
if (vector_y[i] > tmp2) {
tmp2 = vector_y[i];
max_y = vector_y[i];
}
}
printf("vector_x max value %lf\n", max_x);
printf("vector_y max value %lf\n", max_y);
}
static void display_model_line()
{
sprintf(intercept, "%0.7lf", teta_0_intercept);
sprintf(grad, "%0.7lf", teta_1_grad);
sprintf(xrange, "%0.7lf", max_x + 1);
sprintf(yrange, "%0.7lf", max_y + 1);
concatenate(cmd_gnu_1, intercept);
concatenate(cmd_gnu_2, grad);
concatenate(cmd_gnu_3, xrange);
concatenate(cmd_gnu_3, "]");
concatenate(cmd_gnu_4, yrange);
concatenate(cmd_gnu_4, "]");
printf("grad = %s\n", grad);
printf("intercept = %s\n", intercept);
printf("xrange = %s\n", xrange);
printf("yrange = %s\n", yrange);
printf("cmd_gnu_0: %s\n", cmd_gnu_0);
printf("cmd_gnu_1: %s\n", cmd_gnu_1);
printf("cmd_gnu_2: %s\n", cmd_gnu_2);
printf("cmd_gnu_3: %s\n", cmd_gnu_3);
printf("cmd_gnu_4: %s\n", cmd_gnu_4);
printf("cmd_gnu_5: %s\n", cmd_gnu_5);
printf("cmd_gnu_6: %s\n", cmd_gnu_6);
/* print plot */
FILE *gnuplot_pipe = (FILE*)popen("gnuplot -persistent", "w");
FILE *temp = (FILE*)fopen("data.temp", "w");
/* create data.temp */
size_t i;
for (i = 0; i < size; i++)
{
fprintf(temp, "%f %f \n", vector_x[i], vector_y[i]);
}
/* display gnuplot */
for (i = 0; i < 7; i++)
{
fprintf(gnuplot_pipe, "%s \n", commands_gnuplot[i]);
}
}
int main(void)
{
printf("===========================================\n");
printf("INPUT DATA\n");
printf("===========================================\n");
user_input();
display_vector();
printf("\n");
printf("===========================================\n");
printf("COMPUTE MEAN X:Y, TETA_1 TETA_0\n");
printf("===========================================\n");
compute_mean_x_y();
compute_max_x_y();
compute_teta_1_grad();
compute_teta_0_intercept();
printf("\n");
printf("===========================================\n");
printf("COMPUTE LOSS FUNCTION\n");
printf("===========================================\n");
compute_loss_function();
printf("===========================================\n");
printf("COMPUTE R_square\n");
printf("===========================================\n");
compute_r_square(size);
printf("\n");
printf("===========================================\n");
printf("COMPUTE y^ according to x\n");
printf("===========================================\n");
compute_predict_for_x();
printf("\n");
printf("===========================================\n");
printf("DISPLAY LINEAR REGRESSION\n");
printf("===========================================\n");
display_model_line();
printf("\n");
return 0;
}
Look at Section 1 of this paper. This section expresses a 2D linear regression as a matrix multiplication exercise. As long as your data is well-behaved, this technique should permit you to develop a quick least squares fit.
Depending on the size of your data, it might be worthwhile to algebraically reduce the matrix multiplication to simple set of equations, thereby avoiding the need to write a matmult() function. (Be forewarned, this is completely impractical for more than 4 or 5 data points!)
The fastest, most efficient way to solve least squares, as far as I am aware, is to subtract (the gradient)/(the 2nd order gradient) from your parameter vector. (2nd order gradient = i.e. the diagonal of the Hessian.)
Here is the intuition:
Let's say you want to optimize least squares over a single parameter. This is equivalent to finding the vertex of a parabola. Then, for any random initial parameter, x0, the vertex of the loss function is located at x0 - f(1) / f(2). That's because adding - f(1) / f(2) to x will always zero out the derivative, f(1).
Side note: Implementing this in Tensorflow, the solution appeared at w0 - f(1) / f(2) / (number of weights), but I'm not sure if that's due to Tensorflow or if it's due to something else..