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higham_mat_impl.cpp
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higham_mat_impl.cpp
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// Copyright (c) 2020, Massimiliano Fasi and Nicholas J. Higham
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice, this
// list of conditions and the following disclaimer.
//
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
// DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
// FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
// DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
// SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
// CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
// OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
// The above copyright notice and the code are from https://github.com/higham/hpl-ai-matrix
// This file is manual translation of the above software.
#include <math.h>
#include <assert.h>
#include <float.h>
#include <stdio.h>
#define MAX(A,B) ((A)>(B) ? (A): (B))
#define MIN(A,B) ((A)>(B) ? (B): (A))
double fhpl(int n, double alpha, double beta)
{
// compute the inf-norm condition number of the matrix with alpha and beta
// FHPL Value of cond(A,inf) for matrix A(n,a,b).
if(isnan(alpha)) alpha = beta / 2;
double a = alpha, b = beta;
double lambda_1 = 1 + (n-1)*b;
int idash = MIN((int)floor(1./a), n);
int k = MIN((int)floor((1+b)/b), n-1);
double lambda_idash = 1 + (2*k-idash+1)*a + (n-idash)*b + (-k*k+k+3*idash*(idash-1)/2 - n*idash+n)*a*b;
double lambda_n = 1 + (2*k-n+1)*a + (-k*k+k+n*(n-1)/2)*a*b;
double na_est = MAX(MAX(lambda_1, lambda_idash), lambda_n);
double r = (1+a)*(1+b);
int i = 1;
double delta1 = (1+a)*(1./(1+a) + (r==0.?0.:b*(1-pow(r,n-1))/(1-r)));
double deltan = pow(1+a,n) * (1./(1+a));
double ninva_est = MAX(delta1, deltan);
//printf("Z %e %e %e %e %e %e %e %e %e\n", a, b, lambda_1, lambda_idash, lambda_n, delta1, deltan, na_est, ninva_est);
double ret = na_est * ninva_est;
if(isinf(ret)) return DBL_MAX;
else return ret;
}
template<typename F>
double zero_find(F f, double left, double right)
{
// bisection method
// the brent method consumes half # of f evaluations, it's not good enought for complication
double fl = f(left);
double fr = f(right);
if(fl > 0. || fr < 0.) return 0./0.; // nan
while(true){
double tol1 = (2. * fabs(right) + 0.5) * DBL_EPSILON;
if(right-left < tol1) break;
double middle = (left+right)/2;
if(middle==left || middle==right) break;
double fm = f(middle);
//printf("%e %e %e :: %e %e %e\n", left, middle, right, fl, fm, fr);
if(fm==0.) return middle;
else if(fm<0.) {
left = middle;
fl = fm;
}
else {
right = middle;
fr = fm;
}
}
return right;
}
extern "C"
double higham_mat_comp_beta(int n, double kappa, double rho)
{
// % Compute alpha and beta to give cond(A,inf) = kappa.
double left = DBL_EPSILON;
double left_val = fhpl(n, rho*left, left) - kappa;
assert(left_val < 0.);
double right = 1./rho;
int k = 1;
while(true){
double right_val = fhpl(n, rho*right, right) - kappa;
if(isfinite(right_val) && right_val > 0.) break;
// %fprintf('F at right endpoint, right = %9.2e, is %9.2e.\n', right, right_val)
right *= 0.5;
++k;
if(k==100) break;
}
double beta = zero_find([=](double x) -> double {return fhpl(n,rho*x,x)-kappa;}, left, right);
double alpha = rho * beta;
while(alpha > 1.){
// fprintf('Initial alpha = %9.2e exceeds 1 so recomputing.\n', alpha)
right *= 0.5;
right = right/2;
beta = zero_find([=](double x) -> double {return fhpl(n,rho*x,x)-kappa;}, left, right);
alpha = rho * beta;
}
return beta;
}
extern "C"
void hplai_matrix_impl(int n, double* a, int lda, double alpha, double beta)
{
for(int j=0; j<n; ++j){
for(int i=0; i<j; ++i){
a[j*lda+i] = -beta + alpha*beta*i;
}
a[j*lda+j] = 1. + alpha*beta*j;
for(int i=j+1; i<n; ++i){
a[j*lda+i] = -alpha + alpha*beta*j;
}
}
}
extern "C"
void hplai_matrix(int n, double* a, int lda, double kappa)
{
double rho = 0.5;
double beta = higham_mat_comp_beta(n, kappa, rho);
double alpha = rho * beta;
hplai_matrix_impl(n, a, lda, alpha, beta);
}
#if 0
#include <stdio.h>
int main()
{
int n = 10;
double kappa = 1000;
double rho = 0.125;
for(int n=10; n<100000000; n=(n*3)/2){
double beta = comp_beta(n, kappa, rho);
printf("%d %e %e\n", n, beta, rho*beta*beta*n);
}
return 0;
}
#endif