#
# = NDLINAR: multi-linear, multi-parameter least squares fitting
#
# The multi-dimension fitting library NDLINEAR is not included in GSL,
# but is provided as an extension library. This is available at the
# {Patrick Alken's page}[http://ucsu.colorado.edu/~alken/gsl/].
#
# Contents:
# 1. {Introduction}[link:rdoc/ndlinear_rdoc.html#label-Introduction]
# 1. {Class and methods}[link:rdoc/ndlinear_rdoc.html#label-Class+and+Methods]
# 1. {Examples}[link:rdoc/ndlinear_rdoc.html#label-Examples]
#
# == Introduction
# The NDLINEAR extension provides support for general linear least squares
# fitting to data which is a function of more than one variable (multi-linear or
# multi-dimensional least squares fitting). This model has the form where
# <tt>x</tt> is a vector of independent variables, a_i are the fit coefficients,
# and F_i are the basis functions of the fit. This GSL extension computes the
# design matrix X_{ij = F_j(x_i) in the special case that the basis functions
# separate: Here the superscript value j indicates the basis function
# corresponding to the independent variable x_j. The subscripts (i_1, i_2, i_3,
# ...) refer to which basis function to use from the complete set. These
# subscripts are related to the index i in a complex way, which is the main
# problem this extension addresses. The model then becomes where n is the
# dimension of the fit and N_i is the number of basis functions for the variable
# x_i. Computationally, it is easier to supply the individual basis functions
# u^{(j) than the total basis functions F_i(x). However the design matrix X is
# easiest to construct given F_i(x). Therefore the routines below allow the user
# to specify the individual basis functions u^{(j) and then automatically
# construct the design matrix X.
#
#
# == Class and Methods
# ---
# * GSL::MultiFit::Ndlinear.alloc(n_dim, N, u, params)
# * GSL::MultiFit::Ndlinear::Workspace.alloc(n_dim, N, u, params)
#
# Creates a workspace for solving multi-parameter, multi-dimensional linear
# least squares problems. <tt>n_dim</tt> specifies the dimension of the fit
# (the number of independent variables in the model). The array <tt>N</tt> of
# length <tt>n_dim</tt> specifies the number of terms in each sum, so that
# <tt>N[i]</tt>
# specifies the number of terms in the sum of the i-th independent variable.
# The array of <tt>Proc</tt> objects <tt>u</tt> of length <tt>n_dim</tt> specifies
# the basis functions for each independent fit variable, so that <tt>u[i]</tt>
# is a procedure to calculate the basis function for the i-th
# independent variable.
# Each of the procedures <tt>u</tt> takes three block parameters: a point
# <tt>x</tt> at which to evaluate the basis function, an array y of length
# <tt>N[i]</tt> which is filled on output with the basis function values at
# <tt>x</tt> for all i, and a params argument which contains parameters needed
# by the basis function. These parameters are supplied in the <tt>params</tt>
# argument to this method.
#
# Ex)
#
# N_DIM = 3
# N_SUM_R = 10
# N_SUM_THETA = 11
# N_SUM_PHI = 9
#
# basis_r = Proc.new { |r, y, params|
# params.eval(r, y)
# }
#
# basis_theta = Proc.new { |theta, y, params|
# for i in 0...N_SUM_THETA do
# y[i] = GSL::Sf::legendre_Pl(i, Math::cos(theta));
# end
# }
#
# basis_phi = Proc.new { |phi, y, params|
# for i in 0...N_SUM_PHI do
# if i%2 == 0
# y[i] = Math::cos(i*0.5*phi)
# else
# y[i] = Math::sin((i+1.0)*0.5*phi)
# end
# end
# }
#
# N = [N_SUM_R, N_SUM_THETA, N_SUM_PHI]
# u = [basis_r, basis_theta, basis_phi]
#
# bspline = GSL::BSpline.alloc(4, N_SUM_R - 2)
#
# ndlinear = GSL::MultiFit::Ndlinear.alloc(N_DIM, N, u, bspline)
#
# ---
# * GSL::MultiFit::Ndlinear.design(vars, X, w)
# * GSL::MultiFit::Ndlinear.design(vars, w)
# * GSL::MultiFit::Ndlinear::Workspace#design(vars, X)
# * GSL::MultiFit::Ndlinear::Workspace#design(vars)
#
# Construct the least squares design matrix <tt>X</tt> from the input <tt>vars</tt>
# and the previously specified basis functions. vars is a ndata-by-n_dim
# matrix where the ith row specifies the n_dim independent variables for the
# ith observation.
#
# ---
# * GSL::MultiFit::Ndlinear.est(x, c, cov, w)
# * GSL::MultiFit::Ndlinear::Workspace#est(x, c, cov)
#
# After the least squares problem is solved via <tt>GSL::MultiFit::linear</tt>,
# this method can be used to evaluate the model at the data point <tt>x</tt>.
# The coefficient vector <tt>c</tt> and covariance matrix <tt>cov</tt> are
# outputs from <tt>GSL::MultiFit::linear</tt>. The model output value and
# its error [<tt>y, yerr</tt>] are returned as an array.
#
# ---
# * GSL::MultiFit::Ndlinear.calc(x, c, w)
# * GSL::MultiFit::Ndlinear::Workspace#calc(x, c)
#
# This method is similar to <tt>GSL::MultiFit::Ndlinear.est</tt>, but does
# not compute the model error. It computes the model value at the data point
# <tt>x</tt> using the coefficient vector <tt>c</tt> and returns the model
# value.
#
# == Examples
# This example program generates data from the 3D isotropic harmonic oscillator
# wavefunction (real part) and then fits a model to the data using B-splines in
# the r coordinate, Legendre polynomials in theta, and sines/cosines in phi.
# The exact form of the solution is (neglecting the normalization constant for
# simplicity) The example program models psi by default.
#
# #!/usr/bin/env ruby
# require("gsl")
#
# N_DIM = 3
# N_SUM_R = 10
# N_SUM_THETA = 10
# N_SUM_PHI = 9
# R_MAX = 3.0
#
# def psi_real_exact(k, l, m, r, theta, phi)
# rr = GSL::pow(r, l)*Math::exp(-r*r)*GSL::Sf::laguerre_n(k, l + 0.5, 2 * r * r)
# tt = GSL::Sf::legendre_sphPlm(l, m, Math::cos(theta))
# pp = Math::cos(m*phi)
# rr*tt*pp
# end
#
# basis_r = Proc.new { |r, y, params|
# params.eval(r, y)
# }
#
# basis_theta = Proc.new { |theta, y, params|
# for i in 0...N_SUM_THETA do
# y[i] = GSL::Sf::legendre_Pl(i, Math::cos(theta));
# end
# }
#
# basis_phi = Proc.new { |phi, y, params|
# for i in 0...N_SUM_PHI do
# if i%2 == 0
# y[i] = Math::cos(i*0.5*phi)
# else
# y[i] = Math::sin((i+1.0)*0.5*phi)
# end
# end
# }
#
#
# GSL::Rng::env_setup()
#
# k = 5
# l = 4
# m = 2
#
# NDATA = 3000
#
# N = [N_SUM_R, N_SUM_THETA, N_SUM_PHI]
# u = [basis_r, basis_theta, basis_phi]
#
# rng = GSL::Rng.alloc()
#
# bspline = GSL::BSpline.alloc(4, N_SUM_R - 2)
# bspline.knots_uniform(0.0, R_MAX)
#
# ndlinear = GSL::MultiFit::Ndlinear.alloc(N_DIM, N, u, bspline)
# multifit = GSL::MultiFit.alloc(NDATA, ndlinear.n_coeffs)
# vars = GSL::Matrix.alloc(NDATA, N_DIM)
# data = GSL::Vector.alloc(NDATA)
#
#
# for i in 0...NDATA do
# r = rng.uniform()*R_MAX
# theta = rng.uniform()*Math::PI
# phi = rng.uniform()*2*Math::PI
# psi = psi_real_exact(k, l, m, r, theta, phi)
# dpsi = rng.gaussian(0.05*psi)
#
# vars[i][0] = r
# vars[i][1] = theta
# vars[i][2] = phi
#
# data[i] = psi + dpsi
# end
#
# X = GSL::MultiFit::Ndlinear::design(vars, ndlinear)
#
# coeffs, cov, chisq, = GSL::MultiFit::linear(X, data, multifit)
#
# rsq = 1.0 - chisq/data.tss
# STDERR.printf("chisq = %e, Rsq = %f\n", chisq, rsq)
#
# eps_rms = 0.0
# volume = 0.0
# dr = 0.05;
# dtheta = 5.0 * Math::PI / 180.0
# dphi = 5.0 * Math::PI / 180.0
# x = GSL::Vector.alloc(N_DIM)
#
# r = 0.01
# while r < R_MAX do
# theta = 0.0
# while theta < Math::PI do
# phi = 0.0
# while phi < 2*Math::PI do
# dV = r*r*Math::sin(theta)*r*dtheta*dphi
# x[0] = r
# x[1] = theta
# x[2] = phi
#
# psi_model, err = GSL::MultiFit::Ndlinear.calc(x, coeffs, ndlinear)
# psi = psi_real_exact(k, l, m, r, theta, phi)
# err = psi_model - psi
# eps_rms += err * err * dV;
# volume += dV;
#
# if phi == 0.0
# printf("%e %e %e %e\n", r, theta, psi, psi_model)
# end
#
# phi += dphi
# end
# theta += dtheta
# end
# printf("\n");
# r += dr
# end
#
# eps_rms /= volume
# eps_rms = Math::sqrt(eps_rms)
# STDERR.printf("rms error over all parameter space = %e\n", eps_rms)
#
#
# {Reference index}[link:rdoc/ref_rdoc.html]
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#
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