Givens rotations provide a robust and easily parallelizable way to implement QR decomposition. A Givens rotation requires the computation of sine and cosine components of a rotation angle. In the case of real computation, this typically involves the computation of the reciprocal of the hypot() function to normalize a two-vector, as shown for example in Wikipedia.
While this avoids most cases of overflow and underflow in intermediate computation, for very large values a, b, hypot(a,b) may overflow to infinity, while 1/√(a2+b2) is actually representable as a subnormal floating-point number. Also, the use of a division adds further computational cost that can be significant on platforms with slow floating-point division.
A function rhypot(a,b) that directly computes 1/√(a2+b2) at a cost similar to the standard hypot() function would therefore be desirable. The accuracy should be same or better than the naive approach of computing 1.0/hypot(a,b). With a correctly-rounded hypot function, this expression has a maximum error of 1.5 ulps.
How can such a function be implemented efficiently and accurately? The use of IEEE-754 binary floating-point arithmetic and the availability of native hardware support for fused multiply-add (FMA) operations can be assumed. For ease of exposition and testing, we can restrict to single-precision computation, i.e. the IEEE-754 binary32 format.
In the following, I am showing ISO-C99 code that implements rhypot with good accuracy and good performance. The general algorithm is directly derived from the example implementations I showed for hypot in this answer. For hypot, one determines the value of largest magnitude among the arguments, then find a scale factor (a power of two for reasons of accuracy) that maps this value into the vicinity of unity. The scale factor is applied to both arguments, and the length of this transformed 2-vector is then computed with the sqrt function, finally the result scaled back with the "inverse' of the scale factor. The scaling relies on actual multiplication as the arguments may be subnormals that cannot be scaled correctly by simple exponent manipulation alone.
For rhypot, only two changes are needed: the reciprocal square root function rsqrt must be used instead of sqrt, and input scaling and result scaling use the same scale factor.
Some computing environments provide an rsqrt() function, and this function is scheduled for inclusion in a future version of the ISO C standard (ISO/IEC TS 18661-4:2015). For environments that do not provide a reciprocal square root function, I am showing some portable (within the platform requirements stated in the question) and machine-specific implementations.
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>
#include <math.h>
uint32_t __float_as_uint32 (float a)
{
uint32_t r;
memcpy (&r, &a, sizeof r);
return r;
}
float __uint32_as_float (uint32_t a)
{
float r;
memcpy (&r, &a, sizeof r);
return r;
}
float my_rsqrtf (float);
/* Compute the reciprocal of sqrt (a**2 + b**2), avoiding premature overflow
and underflow in intermediate computation. The accuracy of this function
depends on the accuracy of the reciprocal square root implementation used.
With the rsqrtf() implementations shown below, the following maximum ulp
error was observed for 2**36 random test cases:
CORRECTLY_ROUNDED 1.20736973
SSE_HALLEY 1.33120522
SSE_2NR 1.42086841
SQRT_OOX 1.42906701
BIT_TWIDDLE_3NR 1.43062950
ITO_TAKAGI_YAJIMA_1NR 1.43681737
BIT_TWIDDLE_NR_HALLEY 1.47485797
*/
float my_rhypotf (float a, float b)
{
float fa, fb, mn, mx, scale, s, w, res;
uint32_t expo;
/* sort arguments by magnitude */
fa = fabsf (a);
fb = fabsf (b);
mx = fmaxf (fa, fb);
mn = fminf (fa, fb);
/* compute scale factor */
expo = __float_as_uint32 (mx) & 0xfc000000;
scale = __uint32_as_float (0x7e000000 - expo);
/* scale operand of maximum magnitude towards unity */
mn = mn * scale;
mx = mx * scale;
/* mx in [2**-23, 2**6) */
s = fmaf (mx, mx, mn * mn); // 0.75 ulp
w = my_rsqrtf (s);
/* reverse previous scaling */
res = w * scale;
/* handle special cases */
float t = a + b;
if (!(fabsf (t) <= INFINITY)) res = t; // isnan(t)
if (mx == INFINITY) res = 0.0f; // isinf(mx)
return res;
}
#define CORRECTLY_ROUNDED (1)
#define SSE_HALLEY (2)
#define SSE_2NR (3)
#define ITO_TAKAGI_YAJIMA_1NR (4)
#define SQRT_OOX (5)
#define BIT_TWIDDLE_3NR (6)
#define BIT_TWIDDLE_NR_HALLEY (7)
#define RSQRT_VARIANT (SSE_HALLEY)
#if (RSQRT_VARIANT == SSE_2NR) || (RSQRT_VARIANT == SSE_HALLEY)
#include "immintrin.h"
#endif // (RSQRT_VARIANT == SSE_2NR) || (RSQRT_VARIANT == SSE_HALLEY)
float my_rsqrtf (float a)
{
#if RSQRT_VARIANT == CORRECTLY_ROUNDED
float r = (float) sqrt (1.0/(double)a);
#elif RSQRT_VARIANT == SQRT_OOX
float r = sqrtf (1.0f / a);
#elif RSQRT_VARIANT == SSE_2NR
float r;
/* compute initial approximation */
_mm_store_ss (&r, _mm_rsqrt_ss (_mm_set_ss (a)));
/* refine approximation using two Newton-Raphson iterations */
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
#elif RSQRT_VARIANT == SSE_HALLEY
float e, r;
/* compute initial approximation */
_mm_store_ss (&r, _mm_rsqrt_ss (_mm_set_ss (a)));
/* refine approximation using Halley iteration with cubic convergence */
e = fmaf (r * r, -a, 1.0f);
r = fmaf (fmaf (0.375f, e, 0.5f), e * r, r);
#elif RSQRT_VARIANT == BIT_TWIDDLE_3NR
float r;
/* compute initial approximation */
r = __uint32_as_float (0x5f375b0d - (__float_as_uint32(a) >> 1));
/* refine approximation using three Newton-Raphson iterations */
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
#elif RSQRT_VARIANT == BIT_TWIDDLE_NR_HALLEY
float e, r;
/* compute initial approximation */
r = __uint32_as_float (0x5f375b0d - (__float_as_uint32(a) >> 1));
/* refine approximation using Newton-Raphson iteration */
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
/* refine approximation using Halley iteration with cubic convergence */
e = fmaf (r * r, -a, 1.0f);
r = fmaf (fmaf (0.375f, e, 0.5f), e * r, r);
#elif RSQRT_VARIANT == ITO_TAKAGI_YAJIMA_1NR
/* Masayuki Ito, Naofumi Takagi, Shuzo Yajima, "Efficient Initial
Approximation for Multiplicative Division and Square Root by a
Multiplication with Operand Modification". IEEE Transactions on
Computers, Vol. 46, No. 4, April 1997, pp. 495-498.
*/
#define TAB_INDEX_BITS (7)
#define TAB_ENTRY_BITS (16)
#define TAB_ENTRIES (1 << TAB_INDEX_BITS)
#define FP32_EXPO_BIAS (127)
#define FP32_MANT_BITS (23)
#define FP32_SIGN_MASK (0x80000000)
#define FP32_EXPO_MASK (0x7f800000)
#define FP32_EXPO_LSB_MASK (1u << FP32_MANT_BITS)
#define FP32_INDEX_MASK (((1u << TAB_INDEX_BITS) - 1) << (FP32_MANT_BITS - TAB_INDEX_BITS))
#define FP32_XHAT_MASK (~(FP32_INDEX_MASK | FP32_SIGN_MASK) | FP32_EXPO_MASK)
#define FP32_FLIP_BIT_MASK (3u << (FP32_MANT_BITS - TAB_INDEX_BITS - 1))
#define FP32_ONE_HALF (0x3f000000)
const uint16_t d1tab [TAB_ENTRIES] = {
0xb2ec, 0xaed7, 0xaae9, 0xa720, 0xa37b, 0x9ff7, 0x9c93, 0x994d,
0x9623, 0x9316, 0x9022, 0x8d47, 0x8a85, 0x87d8, 0x8542, 0x82c0,
0x8053, 0x7bf0, 0x775f, 0x72f1, 0x6ea4, 0x6a77, 0x666a, 0x6279,
0x5ea5, 0x5aed, 0x574e, 0x53c9, 0x505d, 0x4d07, 0x49c8, 0x469e,
0x438a, 0x408a, 0x3d9e, 0x3ac4, 0x37fc, 0x3546, 0x32a0, 0x300b,
0x2d86, 0x2b10, 0x28a8, 0x264f, 0x2404, 0x21c6, 0x1f95, 0x1d70,
0x1b58, 0x194c, 0x174b, 0x1555, 0x136a, 0x1189, 0x0fb2, 0x0de6,
0x0c22, 0x0a68, 0x08b7, 0x070f, 0x056f, 0x03d8, 0x0249, 0x00c1,
0xfd08, 0xf742, 0xf1b4, 0xec5a, 0xe732, 0xe239, 0xdd6d, 0xd8cc,
0xd454, 0xd002, 0xcbd6, 0xc7cd, 0xc3e5, 0xc01d, 0xbc75, 0xb8e9,
0xb57a, 0xb225, 0xaeeb, 0xabc9, 0xa8be, 0xa5cb, 0xa2ed, 0xa024,
0x9d6f, 0x9ace, 0x983e, 0x95c1, 0x9355, 0x90fa, 0x8eae, 0x8c72,
0x8a45, 0x8825, 0x8614, 0x8410, 0x8219, 0x802e, 0x7c9c, 0x78f5,
0x7565, 0x71eb, 0x6e85, 0x6b31, 0x67f3, 0x64c7, 0x61ae, 0x5ea7,
0x5bb0, 0x58cb, 0x55f6, 0x5330, 0x5079, 0x4dd1, 0x4b38, 0x48ad,
0x462f, 0x43be, 0x4159, 0x3f01, 0x3cb5, 0x3a75, 0x3840, 0x3616
};
uint32_t arg, idx, d1, xhat;
float r;
arg = __float_as_uint32 (a);
idx = (arg >> ((FP32_MANT_BITS + 1) - TAB_INDEX_BITS)) & ((1u << TAB_INDEX_BITS) - 1);
d1 = FP32_ONE_HALF | (d1tab[idx] << ((FP32_MANT_BITS + 1) - TAB_ENTRY_BITS));
xhat = ((arg & FP32_INDEX_MASK) | (((((3 * FP32_EXPO_BIAS) << FP32_MANT_BITS) + ~arg) >> 1) & FP32_XHAT_MASK)) ^ FP32_FLIP_BIT_MASK;
/* compute initial approximation, accurate to about 14 bits */
r = __uint32_as_float (d1) * __uint32_as_float (xhat);
/* refine approximation with one Newton-Raphson iteration */
r = fmaf (fmaf (-a, r * r, 1.0f), 0.5f * r, r);
#else
#error unsupported RSQRT_VARIANT
#endif // RSQRT_VARIANT
return r;
}
uint64_t __double_as_uint64 (double a)
{
uint64_t r;
memcpy (&r, &a, sizeof r);
return r;
}
double floatUlpErr (float res, double ref)
{
uint64_t i, j, err, refi;
int expoRef;
/* ulp error cannot be computed if either operand is NaN, infinity, zero */
if (isnan (res) || isnan (ref) || isinf (res) || isinf (ref) ||
(res == 0.0f) || (ref == 0.0f)) {
return 0.0;
}
/* Convert the float result to an "extended float". This is like a float
with 56 instead of 24 effective mantissa bits.
*/
i = ((uint64_t)__float_as_uint32(res)) << 32;
/* Convert the double reference to an "extended float". If the reference is
>= 2^129, we need to clamp to the maximum "extended float". If reference
is < 2^-126, we need to denormalize because of the float types's limited
exponent range.
*/
refi = __double_as_uint64(ref);
expoRef = (int)(((refi >> 52) & 0x7ff) - 1023);
if (expoRef >= 129) {
j = 0x7fffffffffffffffULL;
} else if (expoRef < -126) {
j = ((refi << 11) | 0x8000000000000000ULL) >> 8;
j = j >> (-(expoRef + 126));
} else {
j = ((refi << 11) & 0x7fffffffffffffffULL) >> 8;
j = j | ((uint64_t)(expoRef + 127) << 55);
}
j = j | (refi & 0x8000000000000000ULL);
err = (i < j) ? (j - i) : (i - j);
return err / 4294967296.0;
}
double rhypot (double a, double b)
{
return 1.0 / hypot (a, b);
}
// Fixes via: Greg Rose, KISS: A Bit Too Simple. http://eprint.iacr.org/2011/007
static unsigned int z=362436069,w=521288629,jsr=362436069,jcong=123456789;
#define znew (z=36969*(z&0xffff)+(z>>16))
#define wnew (w=18000*(w&0xffff)+(w>>16))
#define MWC ((znew<<16)+wnew)
#define SHR3 (jsr^=(jsr<<13),jsr^=(jsr>>17),jsr^=(jsr<<5)) /* 2^32-1 */
#define CONG (jcong=69069*jcong+13579) /* 2^32 */
#define KISS ((MWC^CONG)+SHR3)
#define FP32_QNAN_BIT (0x00400000)
int main (void)
{
float af, bf, resf, reff;
uint32_t ai, bi, resi, refi;
double ref, err, maxerr = 0;
uint64_t diff, diffsum = 0, count = 1ULL << 36;
do {
ai = KISS;
bi = KISS;
af = __uint32_as_float (ai);
bf = __uint32_as_float (bi);
resf = my_rhypotf (af, bf);
ref = rhypot ((double)af, (double)bf);
reff = (float)ref;
refi = __float_as_uint32 (reff);
resi = __float_as_uint32 (resf);
diff = llabs ((long long int)resi - (long long int)refi);
/* If both inputs are a NaN, result can be either argument, converted
to QNaN if necessary. If one input is NaN and the other not infinity
the NaN input must be returned, converted to QNaN if necessary. If
one input is infinity, zero must be returned even if the other input
is a NaN. In all other cases allow up to 1 ulp of difference.
*/
if ((isnan (af) && isnan (bf) && (resi != (ai | FP32_QNAN_BIT)) && (resi != (bi | FP32_QNAN_BIT))) ||
(isnan (af) && !isinf (bf) && !isnan (bf) && (resi != (ai | FP32_QNAN_BIT))) ||
(isnan (bf) && !isinf (af) && !isnan (af) && (resi != (bi | FP32_QNAN_BIT))) ||
(isinf (af) && (resi != 0)) ||
(isinf (bf) && (resi != 0)) ||
(diff > 1)) {
printf ("err # (%08x,%08x): res= %08x (%15.8e) ref=%08x (%15.8e)\n",
ai, bi, resi, resf, refi, reff);
return EXIT_FAILURE;
}
diffsum += diff;
err = floatUlpErr (resf, ref);
if (err > maxerr) {
printf ("ulp=%.8f # (% 15.8e, % 15.8e): res=%15.6a ref=%22.13a\n",
err, af, bf, resf, ref);
maxerr = err;
}
count--;
} while (count);
printf ("diffsum = %llu\n", diffsum);
return EXIT_SUCCESS;
}
I've got an error, regarding calling JacobiSVD in my cuda function.
This is the part of the code that causing the error.
Eigen::JacobiSVD<Eigen::Matrix3d> svd( cov_e, Eigen::ComputeThinU | Eigen::ComputeThinV);
And this is the error message.
CUDA_voxel_building.cu(43): error: calling a __host__
function("Eigen::JacobiSVD , (int)2> ::JacobiSVD") from a __global__
function("kernel") is not allowed
I've used the following command to compile it.
nvcc -std=c++11 -D_MWAITXINTRIN_H_INCLUDED -D__STRICT_ANSI__ -ptx CUDA_voxel_building.cu
I'm using code 8.0 with eigen3 on ubuntu 16.04.
It seems like other functions such as eigen value decomposition also gives the same error.
Anyone knows a solution? I'm enclosing my code below.
//nvcc -ptx CUDA_voxel_building.cu
#include </usr/include/eigen3/Eigen/Core>
#include </usr/include/eigen3/Eigen/SVD>
/*
#include </usr/include/eigen3/Eigen/Sparse>
#include </usr/include/eigen3/Eigen/Dense>
#include </usr/include/eigen3/Eigen/Eigenvalues>
*/
__global__ void kernel(double *p, double *breaks,double *ind, double *mu, double *cov, double *e,double *v, int *n, char *isgood, int minpts, int maxgpu){
bool debuginfo = false;
int idx = threadIdx.x + blockIdx.x * blockDim.x;
if(debuginfo)printf("Thread %d got pointer\n",idx);
if( idx < maxgpu){
int s_ind = breaks[idx];
int e_ind = breaks[idx+1];
int diff = e_ind-s_ind;
if(diff >minpts){
int cnt = 0;
Eigen::MatrixXd local_p(3,diff) ;
for(int k = s_ind;k<e_ind;k++){
int temp_ind=ind[k];
//Eigen::Matrix<double, 3, diff> local_p;
local_p(1,cnt) = p[temp_ind*3];
local_p(2,cnt) = p[temp_ind*3+1];
local_p(3,cnt) = p[temp_ind*3+2];
cnt++;
}
Eigen::Matrix3d centered = local_p.rowwise() - local_p.colwise().mean();
Eigen::Matrix3d cov_e = (centered.adjoint() * centered) / double(local_p.rows() - 1);
Eigen::JacobiSVD<Eigen::Matrix3d> svd( cov_e, Eigen::ComputeThinU | Eigen::ComputeThinV);
/* Eigen::Matrix3d Cp = svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().transpose();
mu[idx]=p[ind[s_ind]*3];
mu[idx+1]=p[ind[s_ind+1]*3];
mu[idx+2]=p[ind[s_ind+2]*3];
e[idx]=svd.singularValues()(0);
e[idx+1]=svd.singularValues()(1);
e[idx+2]=svd.singularValues()(2);
n[idx] = diff;
isgood[idx] = 1;
for(int x = 0; x < 3; x++)
{
for(int y = 0; y < 3; y++)
{
v[x+ 3*y +idx*9]=svd.matrixV()(x, y);
cov[x+ 3*y +idx*9]=cov_e(x, y);
//if(debuginfo)printf("%f ",R[x+ 3*y +i*9]);
if(debuginfo)printf("%f ",Rm(x, y));
}
}
*/
} else {
mu[idx]=0;
mu[idx+1]=0;
mu[idx+2]=0;
e[idx]=0;
e[idx+1]=0;
e[idx+2]=0;
n[idx] = 0;
isgood[idx] = 0;
for(int x = 0; x < 3; x++)
{
for(int y = 0; y < 3; y++)
{
v[x+ 3*y +idx*9]=0;
cov[x+ 3*y +idx*9]=0;
}
}
}
}
}
First of all, Ubuntu 16.04 provides Eigen 3.3-beta1, which is not really recommended to be used. I would suggest upgrading to a more recent version. Furthermore, to include Eigen, write (e.g.):
#include <Eigen/Eigenvalues>
and compile with -I /usr/include/eigen3 (if you use the version provided by the OS), or better -I /path/to/local/eigen-version.
Then, as talonmies noted, you can't call host-functions from kernels, (I'm not sure at the moment, why JacobiSVD is not marked as device function), but in your case it would make much more sense to use Eigen::SelfAdjointEigenSolver, anyway. Since the matrix you are decomposing is fixed-size 3x3 you should actually use the optimized computeDirect method:
Eigen::SelfAdjointEigenSolver<Eigen::Matrix3d> eig; // default constructor
eig.computeDirect(cov_e); // works for 2x2 and 3x3 matrices, does not require loops
It seems the computeDirect even works on the beta version provided by Ubuntu (I'd still recommend to update).
Some unrelated notes:
The following is wrong, since you should start with index 0:
local_p(1,cnt) = p[temp_ind*3];
local_p(2,cnt) = p[temp_ind*3+1];
local_p(3,cnt) = p[temp_ind*3+2];
Also, you can write this in one line:
local_p.col(cnt) = Eigen::Vector3d::Map(p+temp_ind*3);
This line will not fit (unless diff==3):
Eigen::Matrix3d centered = local_p.rowwise() - local_p.colwise().mean();
What you probably mean is (local_p is actually 3xn not nx3)
Eigen::Matrix<double, 3, Eigen::Dynamic> centered = local_p.colwise() - local_p.rowwise().mean();
And when computing cov_e you need to .adjoint() the second factor, not the first.
You can avoid both 'big' matrices local_p and centered, by directly accumulating Eigen::Matrix3d sum2 and Eigen::Vector3d sum with sum2 += v*v.adjoint() and sum +=v and computing
Eigen::Vector3d mu = sum / diff;
Eigen::Matrix3d cov_e = (sum2 - mu*mu.adjoint()*diff)/(diff-1);
I visited the gnu gsl website and i dont find the example there to solve a differential equation to be intuitive at all (especially because it is using 2nd order differential equation). https://www.gnu.org/software/gsl/manual/html_node/ODE-Example-programs.html#ODE-Example-programs
Can somebody guide about where to find a descriptive guide to how solve a very simple first order differetial equation.
For example, supoose my function is y'=x+2y (or any such function) then how do i write code in gsl to solve it with a given fixed step size and initial condition.
For y'=f(x,y)=x+2y the arrays have all dimension 1, which normally is something to avoid, but here it is instructional. For the explicit solvers, i.e., those not containing imp in the name, you do not need the Jacobian:
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv2.h>
int odefunc (double x, const double y[], double f[], void *params)
{
f[0] = x+2*y[0];
return GSL_SUCCESS;
}
int * jac;
int main ()
{
int dim = 1;
gsl_odeiv2_system sys = {odefunc, NULL, dim, NULL};
gsl_odeiv2_driver * d = gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rkf45, 1e-6, 1e-6, 0.0);
int i;
double x0 = 0.0, xf = 100.0; /* start and end of integration interval */
double x = x0;
double y[1] = { 1.0 }; /* initial value */
for (i = 1; i <= 100; i++)
{
double xi = x0 + i * (xf-x0) / 100.0;
int status = gsl_odeiv2_driver_apply (d, &x, xi, y);
if (status != GSL_SUCCESS)
{
printf ("error, return value=%d\n", status);
break;
}
printf ("%.8e %.8e\n", x, y[0]);
}
gsl_odeiv2_driver_free (d);
return 0;
}
You may want to look up the book "Introduction to Computational Modeling Using C and Open-Source Tools" by Jose M. Garrido.
Lutzl, Please review:
'#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv2.h>
int odefunc (double x, const double y[], double f[], void *params)
{
f[0] = x+2*y[0];
return GSL_SUCCESS;
}
int jac(double x , const double y[] ,double *dfdy , double dfdx[], void *params) {
gsl_matrix_view dfdy_mat= gsl_matrix_view_array(dfdy,1,1);
gsl_matrix *m= &dfdy_mat.matrix;
gsl_matrix_set(m,0,0,x);
dfdx[0]=2;
return GSL_SUCCESS;
}
int main ()
{
int dim =1;
gsl_odeiv2_system sys = {odefunc, jac, dim, NULL};
gsl_odeiv2_driver * d = gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk1imp,1e-7,1e-7, 0.0);
int i;
double x0 = 0.0, xf =1.0; /*al value */
while(x<xf)
{
double xi = x0 + 0.25;
int status = gsl_odeiv2_driver_apply (d, &x, xi, y);
if (status != GSL_SUCCESS)
{
printf ("error, return value=%d\n", status);
break;
}
printf ("%.8e %.8e\n", x, y[0]);
}
gsl_odeiv2_driver_free (d);
return 0;
}
'
i am trying to write a code to display Mandelbrot set for the numbers between
(-3,-3) to (2,2) on my terminal.
The main function generates & feeds a complex number to analyze function.
The analyze function returns character "*" for the complex number Z within the set and "." for the numbers which lie outside the set.
The code:
#define MAX_A 2 // upperbound on real
#define MAX_B 2 // upper bound on imaginary
#define MIN_A -3 // lowerbnd on real
#define MIN_B -3 // lower bound on imaginary
#define NX 300 // no. of points along x
#define NY 200 // no. of points along y
#define max_its 50
int analyze(double real,double imag);
void main()
{
double a,b;
int x,x_arr,y,y_arr;
int array[NX][NY];
int res;
for(y=NY-1,x_arr=0;y>=0;y--,x_arr++)
{
for(x=0,y_arr++;x<=NX-1;x++,y_arr++)
{
a= MIN_A+ ( x/( (double)NX-1)*(MAX_A-MIN_A) );
b= MIN_B+ ( y/( (double)NY-1 )*(MAX_B-MIN_B) );
//printf("%f+i%f ",a,b);
res=analyze(a,b);
if(res>49)
array[x][y]=42;
else
array[x][y]=46;
}
// printf("\n");
}
for(y=0;y<NY;y++)
{
for(x=0;x<NX;x++)
printf("%2c",array[x][y]);
printf("\n");
}
}
The analyze function accepts the coordinate on imaginary plane ;
and computes (Z^2)+Z 50 times ; and while computing if the complex number explodes, then function returns immidiately else the function returns after finishing 50 iterations;
int analyze(double real,double imag)
{
int iter=0;
double r=4.0;
while(iter<50)
{
if ( r < ( (real*real) + (imag*imag) ) )
{
return iter;
}
real= ( (real*real) - (imag*imag) + real);
imag= ( (2*real*imag)+ imag);
iter++;
}
return iter;
}
So, i am analyzing 60000 (NX * NY) numbers & displaying it on the terminal
considering 3:2 ratio (300,200) , i even tried 4:3 (NX:NY) , but the output remains same and the generated shape is not even close to the mandlebrot set :
hence, the output appears inverted ,
i browsed & came across lines like:
(x - 400) / ZOOM;
(y - 300) / ZOOM;
on many mandelbrot codes , but i am unable to understand how this line may rectify my output.
i guess i am having trouble in mapping output to the terminal!
(LB_Real,UB_Imag) --- (UB_Real,UB_Imag)
| |
(LB_Real,LB_Imag) --- (UB_Real,LB_Imag)
Any Hint/help will be very useful
The Mandelbrot recurrence is zn+1 = zn2 + c.
Here's your implementation:
real= ( (real*real) - (imag*imag) + real);
imag= ( (2*real*imag)+ imag);
Problem 1. You're updating real to its next value before you've used the old value to compute the new imag.
Problem 2. Assuming you fix problem 1, you're computing zn+1 = zn2 + zn.
Here's how I'd do it using double:
int analyze(double cr, double ci) {
double zr = 0, zi = 0;
int r;
for (r = 0; (r < 50) && (zr*zr + zi*zi < 4.0); ++r) {
double zr1 = zr*zr - zi*zi + cr;
double zi1 = 2 * zr * zi + ci;
zr = zr1;
zi = zi1;
}
return r;
}
But it's easier to understand if you use the standard C99 support for complex numbers:
#include <complex.h>
int analyze(double cr, double ci) {
double complex c = cr + ci * I;
double complex z = 0;
int r;
for (r = 0; (r < 50) && (cabs(z) < 2); ++r) {
z = z * z + c;
}
return r;
}
I have to write in a PyCUDA function that gets two matrices Nx3 and Mx3, and return a matrix NxM, but I can't figure out how to pass by reference a matrix without knowing the number of columns.
My code basically is something like that:
#kernel declaration
mod = SourceModule("""
__global__ void distance(int N, int M, float d1[][3], float d2[][3], float res[][M])
{
int i = threadIdx.x;
int j = threadIdx.y;
float x, y, z;
x = d2[j][0]-d1[i][0];
y = d2[j][1]-d1[i][1];
z = d2[j][2]-d1[i][2];
res[i][j] = x*x + y*y + z*z;
}
""")
#load data
data1 = numpy.loadtxt("data1.txt").astype(numpy.float32) # Nx3 matrix
data2 = numpy.loadtxt("data2.txt").astype(numpy.float32) # Mx3 matrix
N=data1.shape[0]
M=data2.shape[0]
res = numpy.zeros([N,M]).astype(numpy.float32) # NxM matrix
#invoke kernel
dist_gpu = mod.get_function("distance")
dist_gpu(cuda.In(numpy.int32(N)), cuda.In(numpy.int32(M)), cuda.In(data1), cuda.In(data2), cuda.Out(res), block=(N,M,1))
#save data
numpy.savetxt("results.txt", res)
Compiling this I receive an error:
kernel.cu(3): error: a parameter is not allowed
that is, I cannot use M as the number of columns for res[][] in the declaretion of the function. I cannot either left the number of columns undeclared...
I need a matrix NxM as an output, but I can't figure out how to do this. Can you help me?
You should use pitched linear memory access inside the kernel, that is how ndarray and gpuarray store data internally, and PyCUDA will pass a pointer to the data in gpu memory allocated for a gpuarray when it is supplied as a argument to a PyCUDA kernel. So (if I understand what you are trying to do) your kernel should be written as something like:
__device__ unsigned int idx2d(int i, int j, int lda)
{
return j + i*lda;
}
__global__ void distance(int N, int M, float *d1, float *d2, float *res)
{
int i = threadIdx.x + blockDim.x * blockIdx.x;
int j = threadIdx.y + blockDim.y * blockIdx.y;
float x, y, z;
x = d2[idx2d(j,0,3)]-d1[idx2d(i,0,3)];
y = d2[idx2d(j,1,3)]-d1[idx2d(i,1,3)];
z = d2[idx2d(j,2,3)]-d1[idx2d(i,2,3)];
res[idx2d(i,j,N)] = x*x + y*y + z*z;
}
Here I have assumed the numpy default row major ordering in defining the idx2d helper function. There are still problems with the Python side of the code you posted, but I guess you know that already.
EDIT: Here is a complete working repro case based of the code posted in your question. Note that it only uses a single block (like the original), so be mindful of block and grid dimensions when trying to run it on anything other than trivially small cases.
import numpy as np
from pycuda import compiler, driver
from pycuda import autoinit
#kernel declaration
mod = compiler.SourceModule("""
__device__ unsigned int idx2d(int i, int j, int lda)
{
return j + i*lda;
}
__global__ void distance(int N, int M, float *d1, float *d2, float *res)
{
int i = threadIdx.x + blockDim.x * blockIdx.x;
int j = threadIdx.y + blockDim.y * blockIdx.y;
float x, y, z;
x = d2[idx2d(j,0,3)]-d1[idx2d(i,0,3)];
y = d2[idx2d(j,1,3)]-d1[idx2d(i,1,3)];
z = d2[idx2d(j,2,3)]-d1[idx2d(i,2,3)];
res[idx2d(i,j,N)] = x*x + y*y + z*z;
}
""")
#make data
data1 = np.random.uniform(size=18).astype(np.float32).reshape(-1,3)
data2 = np.random.uniform(size=12).astype(np.float32).reshape(-1,3)
N=data1.shape[0]
M=data2.shape[0]
res = np.zeros([N,M]).astype(np.float32) # NxM matrix
#invoke kernel
dist_gpu = mod.get_function("distance")
dist_gpu(np.int32(N), np.int32(M), driver.In(data1), driver.In(data2), \
driver.Out(res), block=(N,M,1), grid=(1,1))
print res