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generate_span_exp.m
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generate_span_exp.m
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%% generate_span_exp
%
% Description:
% Generates a basis of F as well as a spanning set of G=(FF)', where F is an exponential approximation space, and the corresponding moments
%
% Author: Jan Glaubitz
% Date: Jan 07, 2022
%
% INPUT:
% x_L, x_R : domain boundaries
% K : dimension
%
% OUTPUT:
% basis_F : vector-valued function with basis of F
% dx_basis_F : vector-valued function with derivatives of the basis of F
% span_G : vector-valued function with spanning elements of G
% m_G : moments corrsponding to G
function [ basis_F, dx_basis_F, span_G, m_G ] = generate_span_exp( x_L, x_R, K )
%% Especially simple special cases
if K == 1
error('K needs to be larger than 1!')
elseif K == 2
basis_F = @(x) [ x.^0; exp(x) ];
dx_basis_F = @(x) [ 0*x; exp(x) ];
elseif K == 3
basis_F = @(x) [ x.^0; x; exp(x) ];
dx_basis_F = @(x) [ 0*x; x.^0; exp(x) ];
%% all other cases
else
%% basis for F and F'
beta = (0:K-2)'; % exponents
basis_F = @(x) [ x.^beta; exp(x) ]; % monomial basis
dx_basis_F = @(x) [ 0*x; x.^0; beta(3:end).*x.^(beta(3:end)-1); exp(x) ]; % monomial basis
end
%% spanning set for G and corresponding moments
span_G = @(x) basis_F(x)*(dx_basis_F(x)') + dx_basis_F(x)*(basis_F(x)');
m_G = basis_F(x_R)*(basis_F(x_R)') - basis_F(x_L)*(basis_F(x_L)');
%% vectorize
span_G = @(x) reshape( span_G(x), [K^2, 1]);
m_G = reshape( m_G, [K^2, 1]);
end