COEFFS_COMPOSITION

Contents

function [Hk] = coeffs_composition(W_0,H,multi_input)
        

Composition coefficients for power series

Calculates the composition coefficients of power series as given in https://doi.org/10.1016/j.jsv.2020.115640 , Appendix C

Take the expansion of row $i$ of a vector valued function $(\mathbf{S(p)} )_i= \sum_{\mathbf{u}}S_{\mathbf{u},i}\mathbf{p^u}$ .

If this power series is multiplied with itself $s$ times, it can be written like $((\mathbf{S(p)} )_i)^s= \bigg( \ \sum_{\mathbf{u}}S_{\mathbf{u},i}\mathbf{p^u}\bigg)^s= \sum_{\mathbf{h}}H_{i,s,\mathbf{h}}\mathbf{p^h}$

This function computes $H_{i,s,\mathbf{k}_f} = \frac{s}{k_{f,j}} \sum_{ \mathbf{u < k}_f} u_j S_{i,\mathbf{u}} H_{i,s-1,\mathbf{k}_f-\mathbf{u}} $ for all multi-index vectors $\mathbf{k}_f$ of a given order $k$ .

[Hk] = COEFFS_COMPOSITION(W_0,H,multi_input)

See also: COHOMOLOGICAL_SOLUTION, FNL_INTRUSIVE

switch multi_input.ordering
    case 'conjugate'
        %Coefficients in conjugate ordering, used in cohomological solution
        %for computing autonomous coefficients
        Hk = conjugate_computation(W_0,H,multi_input);
    case 'revlex'
        %Restoring coefficients for nonautonomous calculations given the
        %calculated SSM coefficients.
        Hk = rev_lexicographic_computation(W_0,H,multi_input.k);
    case 'lex'
        Hk = lexicographic_computation(W_0,H,multi_input.k);
end
        

end

function [Hk] = conjugate_computation(W_0,H,multi_input)
        

Algorithm for unique multi-indices up to conjugation

This function computes the coefficients for the composition of a multivariate polynomial power series.

Take the expansion of row $i$ of a vector valued function $(\mathbf{S(p)} )_i= \sum_{\mathbf{u}}S_{\mathbf{u},i}\mathbf{p^u}$ .

If this power series is multiplied with itself $s$ times, it can be written like $((\mathbf{S(p)} )_i)^s= \bigg( \ \sum_{\mathbf{u}}S_{\mathbf{u},i}\mathbf{p^u}\bigg)^s= \sum_{\mathbf{h}}H_{i,s,\mathbf{h}}\mathbf{p^h}$

This function computes $H_{i,s,\mathbf{k}_f} = \frac{s}{k_{f,j}} \sum_{ \mathbf{u < k}_f} u_j S_{i,\mathbf{u}} H_{i,s-1,\mathbf{k}_f-\mathbf{u}}$ for all multi-index vectors $\mathbf{k}_f$ of a given order $k$ .

Z_cci = multi_input.Z_cci;
N = multi_input.N;
K = multi_input.K;
revlex2conj = multi_input.revlex2conj;
l = size(K,1);
k = sum(K(:,1),1);
string = 'conjugate';
z_k    = Z_cci(k);

Hk = zeros(N,z_k,k);
[kj,Ikj] = min(K +(k+1)*(K ==0)); %find nonzero minima of all multi-indices of order k
        

Loop over all orders of multi-indices that are potentially nonzero and do not require the knowledge of the order $k$ coefficients of the SSM parametrisation.

for ord = 1:k-1
        

Create all multi-indices of order $k-u$

    g       = flip(sortrows(nsumk(l,k-ord,'nonnegative')).',2);

    g       = g(:,revlex2conj{k-ord});

    [gmink,I_k,I_g] = multi_subtraction(K,g,'Arbitrary');
        

Now all the positions of the order $m$ multi-indices are calculated. Assume the vector $\texttt{gmink}$ has $u_\alpha$ columns.

    I_gmk     = multi_index_2_ordering(gmink,string,revlex2conj);
        

We read out all the values $u_j$ positions of minimal entries

    pos     = Ikj(I_k);                 %find row for every vector in mv
    sz      = size(I_k,2);
    poslin  = sub2ind([l,sz],pos,1:sz); %create linear index array
    % read out minimal entries
    gmk_j     = gmink(poslin);            %has to be a row vector
        

One thing that is redundant is all the calculations where $u_j$ is zero. The corresponding values could be ignored, by taking the indices corresponding to $u_j$ = 0 out of calculations .

Consequently, the coefficients $u_jS_{i,\mathbf{u}} H_{i,s-1,\mathbf{k}_f-\mathbf{u}}$ are calculated. In the array Hk the first row corresponds to the phase space dimension, the third one to the scalar exponents.

$\texttt{H}_{k-u}(:,\texttt{I}_{g},:)$ is a matrix in $\mathbf{C}^{2n \times \texttt{sz} \times k-1}$ . Contribution of coefficients where $s=1$ explicitly depend on the SSM coefficients of the order that is calculated and will be added in the end after the highest order coefficients of the SSM parametrisation have been computed,but fortunately are not needed for the calculations of themselves. Finally $\texttt{S\{ord\}}(:,\texttt{gmk\_I}) \in \mathbf{C}^{2n \times \texttt{sz} }$ .

Now, since we want to make use of the symmetry in the SSM coefficients, we split all contributions such that in conjugate multi-index ordering, all of the indices at order $\texttt{ord}$ that are smaller or equal than $\texttt{Z(ord)}$ are read out normally, and the ones that are bigger are changed to their conjugate counterpart.

    coeff       = zeros(N,sz,k-ord);

    z_o   = nchoosek(ord+l-1,l-1);
    z_ko  = nchoosek(k-ord+l-1,l-1);
    z_cci_o = Z_cci(ord);
    z_cci_ko = Z_cci(k-ord);

    % determine which coefficients correspond to multi-indices bigger
    % than the conjugate center index at order o and k-o.

    I_g_a = I_g <= z_cci_ko;
    I_gmk_a = I_gmk <= z_cci_o;
    % where g_I <= z_ko_2, and gmk_I <= z_k_2
    aa = I_g_a & I_gmk_a;
    % where g_I > z_ko_2, but gmk_I <= z_k_2
    ab = ~I_g_a & I_gmk_a;
    % where g_I <= z_ko_2, but gmk_I > z_k_2
    ba = I_g_a & ~I_gmk_a;
    % where g_I > z_ko_2, and gmk_I > z_k_2
    bb = ~ I_g_a & ~I_gmk_a;

    %read out contributions using symmetries
    W0_coeffs = W_0(ord).coeffs;

    coeff(:,aa,:)   = W0_coeffs(:,I_gmk(aa)).* gmk_j(aa)             .* H{k-ord}(:,I_g(aa),:);
    coeff(:,ab,:)   = W0_coeffs(:,I_gmk(ab)).* gmk_j(ab)             .* conj(H{k-ord}(:,z_ko-I_g(ab)+1,:));
    coeff(:,ba,:)   = conj(W0_coeffs(:,z_o-I_gmk(ba)+1)).* gmk_j(ba) .* H{k-ord}(:,I_g(ba),:);
    coeff(:,bb,:)   = conj(W0_coeffs(:,z_o-I_gmk(bb)+1)).* gmk_j(bb) .* conj(H{k-ord}(:,z_ko-I_g(bb)+1,:));
        

$\texttt{coeff} \in \mathbf{C}^{2n \times \texttt{sz} \times k-1}$ . What is left now is to add up all the parts corresponding to a specific multi-index of order $k$ and multiply with the $s$ the slice corresponds to. We want to add all the columns corresponding to where $\texttt{I}_{k}$ has the same values (the columns contain coefficients that contribute to the same order $k$ multi-index).

    for f = unique(I_k)
        Hk(:,f,2:(k-ord+1)) = Hk(:,f,2:(k-ord+1)) + sum(coeff(:,I_k==f,:),2);
    end
        

end
%multiplication with s and kj
s  = 1:k;
%match dimensions for pointiwse multiplication
Hk = permute(Hk,[2,3,1]) .* ((1./kj).' * s);
Hk = permute(Hk,[3,1,2]);
        

end


function [Hk] = lexicographic_computation(W_0,H,k)
        

l = size(H{1},2);
z_k = nchoosek(k+l-1,l-1);
N = size(H{1},1);
K = sortrows(nsumk(l,k,'nonnegative')).';

string = 'lex';
Hk = zeros(N,z_k,k);

%find nonzero minima of all multi-indices of order k
[kj,Ikj] = min(K +(k+1)*(K ==0));
        

Loop over all orders of multi-indices that are potentially nonzero and do not require the knowledge of the order $k$ coefficients of the SSM parametrisation.

for ord = 1:k-1
        

    if isempty(W_0{ord}.coeffs)
        continue
    end
        

Create all multi-indices of order $k-u$

    if l >1
        g       = sortrows(nsumk(l,k-ord,'nonnegative')).'; %lexicographic!!
    else
        g = k-ord;
    end
    [gmink,I_k,I_g] = multi_subtraction(K,g,'Arbitrary');
        

Now all the positions of the order $m$ multi-indices are calculated. Assume the vector $\texttt{gmink}$ has $u_\alpha$ columns.

    I_gmk     = multi_index_2_ordering(gmink,string,[]);
        

We read out all the values $u_j$ positions of minimal entries

    pos     = Ikj(I_k);                 %find row for every vector in mv
    sz      = size(I_k,2);
    poslin  = sub2ind([l,sz],pos,1:sz); %create linear index array
    % read out minimal entries
    gmk_j     = gmink(poslin);            %has to be a row vector
        

One thing that is redundant is all the calculations where $u_j$ is zero. The corresponding values could be ignored, by taking the indices corresponding to $u_j$ = 0 out of calculations .

Consequently, the coefficients $u_jS_{i,\mathbf{u}} H_{i,s-1,\mathbf{k}_f-\mathbf{u}} $ are calculated. In the array $\texttt{Hk}$ the first row corresponds to the phase space dimension, the third one to the scalar exponents.

$\texttt{H}_{k-u}(:,\texttt{I}_{g},:)$ is a matrix in $\mathbf{C}^{2n \times \texttt{sz} \times k-1}$ . Contribution of coefficients where $s=1$ explicitly depend on the SSM coefficients of the order that is calculated and will be added in the end after the highest order coefficients of the SSM parametrisation have been computed,but fortunately are not needed for the calculations of themselves. Finally $\texttt{S\{ord\}}(:,\texttt{gmk\_I}) \in \mathbf{C}^{2n \times \texttt{sz} }$ .

    %read out contributions

    coeff   = W_0(ord).coeffs(:,I_gmk).* gmk_j .* H{k-ord}(:,I_g,:);
        

$\texttt{coeff} \in \mathbf{C}^{2n \times \texttt{sz} \times k-1}$ . What is left now is to add up all the parts corresponding to a specific multi-index of order $k$ and multiply with the $s$ the slice corresponds to. We want to add all the columns corresponding to where $\texttt{I}_{k}$ has the same values (the columns contain coefficients that contribute to the same order $k$ multi-index).

    for f = unique(I_k)
        Hk(:,f,2:(k-ord+1)) = Hk(:,f,2:(k-ord+1)) + sum(coeff(:,I_k==f,:),2);
    end
        

end
%multiplication with s and kj
s  = 1:k;
%match dimensions for pointiwse multiplication
Hk = permute(Hk,[2,3,1]) .* ((1./kj).' * s);
Hk = permute(Hk,[3,1,2]);

% Since we reconstruct we already know the SSM coefficients at that order
if ~isempty(W_0{k}) && ~isempty(W_0{k}.coeffs)
    Hk(:,:,1) = W_0(k).coeffs;
end
        

end


function [Hk] = rev_lexicographic_computation(W_0,H,k)
        

General Algorithm

l = size(H{1},2);
z_k = nchoosek(k+l-1,l-1);
N = size(H{1},1);
K = flip(sortrows(nsumk(l,k,'nonnegative')).',2);

string = 'revlex';
Hk = zeros(N,z_k,k);

%find nonzero minima of all multi-indices of order k
[kj,Ikj] = min(K +(k+1)*(K ==0));
        

Loop over all orders of multi-indices that are potentially nonzero and do not require the knowledge of the order $k$ coefficients of the SSM parametrisation.

for ord = 1:k-1
        

    if isempty(W_0(ord).coeffs)
        continue
    end
        

Create all multi-indices of order $k-u$

    if l > 1
        g       = flip(sortrows(nsumk(l,k-ord,'nonnegative')).',2); %rev-lexicographic!!
    else
        g = k-ord;
    end
    [gmink,I_k,I_g] = multi_subtraction(K,g,'Arbitrary');
        

Now all the positions of the order $m$ multi-indices are calculated. Assume the vector $\texttt{gmink}$ has $u_\alpha$ columns.

    I_gmk     = multi_index_2_ordering(gmink,string,[]);
        

We read out all the values $u_j$ positions of minimal entries

    pos     = Ikj(I_k);                 %find row for every vector in mv
    sz      = size(I_k,2);
    poslin  = sub2ind([l,sz],pos,1:sz); %create linear index array
    % read out minimal entries
    gmk_j     = gmink(poslin);            %has to be a row vector
        

One thing that is redundant is all the calculations where $u_j$ is zero. The corresponding values could be ignored, by taking the indices corresponding to $u_j$ = 0 out of calculations .

Consequently, the coefficients $u_jS_{i,\mathbf{u}} H_{i,s-1,\mathbf{k}_f-\mathbf{u}} $ are calculated. In the array $\texttt{Hk}$ the first row corresponds to the phase space dimension, the third one to the scalar exponents.

$\texttt{H}_{k-u}(:,\texttt{I}_{g},:)$ is a matrix in $\mathbf{C}^{2n \times \texttt{sz} \times k-1}$ . Contribution of coefficients where $s=1$ explicitly depend on the SSM coefficients of the order that is calculated and will be added in the end after the highest order coefficients of the SSM parametrisation have been computed,but fortunately are not needed for the calculations of themselves. Finally $\texttt{S\{ord\}}(:,\texttt{gmk\_I}) \in \mathbf{C}^{2n \times \texttt{sz} }$ .

    %read out contributions

    coeff   = W_0(ord).coeffs(:,I_gmk).* gmk_j .* H{k-ord}(:,I_g,:);
        

$\texttt{coeff} \in \mathbf{C}^{2n \times \texttt{sz} \times k-1}$ . What is left now is to add up all the parts corresponding to a specific multi-index of order $k$ and multiply with the $s$ the slice corresponds to. We want to add all the columns corresponding to where $\texttt{I}_{k}$ has the same values (the columns contain coefficients that contribute to the same order $k$ multi-index).

    for f = unique(I_k)
        Hk(:,f,2:(k-ord+1)) = Hk(:,f,2:(k-ord+1)) + sum(coeff(:,I_k==f,:),2);
    end
        

end
%multiplication with s and kj
s  = 1:k;
%match dimensions for pointiwse multiplication
Hk = permute(Hk,[2,3,1]) .* ((1./kj).' * s);
Hk = permute(Hk,[3,1,2]);

% Since we reconstruct we already know the SSM coefficients at that order
if ~isempty(W_0(k)) && ~isempty(W_0(k).coeffs)
    Hk(:,:,1) = W_0(k).coeffs;
end
        

end
        

Published with MATLAB® R2023a