>> help make_matrices_for_compact_recursions
 
   This function makes the matrices that are needed to state the recurrence
   relations used in the various band algorithms in compact form and adds them
   to the structure "result".
 
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   Input:  A structure "result" generated by a call to one of the functions
           "band_Arnoldi", "band_Lanczos", "sym_band_Lanczos", 
           "Herm_band_Lanczos", "Jsym_band_Lanczos", "JHerm_band_Lanczos",
           which implement the band Arnoldi process and the band Lanczos method
           and its variants
 
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   On return, the structure "result" contains all the fields (with unchanged
   values) of the input structure "result" and the following new fields:
 
      When the input structure "result" was produced by the function
      "band_Arnoldi", the 4 fields
 
            result.H_s,  result.rho_s,  result.V_dh,  result.R_dh
 
      are added to "result".
   
      When the input structure "result" was produced by the function
      "band_Lanczos", the 8 fields
 
            result.T_s,  result.rho_s,  result.V_dh,  result.R_dh
 
      and
 
            result.Tt_s,  result.eta_s,  result.W_dh,  result.L_dh
  
      are added to "result".
 
      When the input structure "result" was produced by any of the functions
      "sym_band_Lanczos", "Herm_band_Lanczos", "Jsym_band_Lanczos", or
      "JHerm_band_Lanczos", the 4 fields
 
            result.T_s,  result.rho_s,  result.V_dh,  result.R_dh
 
      are added to "result".
 
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   Notes:  1) The various band algorithms construct basis vectors for the n-th
              right block Krylov subspace (induced by A and R) by means of
              recurrence relations that can be stated in compact form as
              follows:
 
                   A V = V H_s + V_dh     (for the band Arnoldi process)
 
                   A V = V T_s + V_dh     (for the band Lanczos method and
                                                              its variants)
 
                     R = V(:,1:mt) rho_s + R_dh
 
              Here, V is the matrix the columns of which are the n constructed
              basis vectors and mt = min(m,n).  The matrices H_s, T_s, and 
              rho_s contain the recurrence coefficients, and V_dh and R_dh are
              sparse matrices the nonzero columns of which are the candidate
              vectors and, if deflations have occurred, the deflated vectors.
 
           2) The band Lanzcos method also constructs basis vectors for the
              n-th left block Krylov subspace (induced by A^T and L) by means
              of recurrence relations that can be stated in compact form as
              follows:
 
                   A^T W = W Tt_s + W_dh
 
                       L = W(:,1:pt) eta_s + L_dh	    
 
              Here, W is the matrix the columns of which are the n constructed
              basis vectors and pt = min(p,n).  The matrices Tt_s and eta_s
              contain the recurrence coefficients, and W_dh and L_dh are sparse
              matrices the nonzero columns of which are the candidate vectors
 	      and, if deflations have occurred, the deflated vectors.
 
           3) The matrices H_s, T_s, Tt_s, rho_s, and eta_s are readily 
              obtained by zeroing out the vertical spikes (due to possible
 	      deflations) of the matrices H, T, Tt, rho, and eta that are
              generated by the functions "band_Arnoldi", "band_Lanczos",
 	      "sym_band_Lanczos", "Herm_band_Lanczos", "Jsym_band_Lanczos", or
              "JHerm_band_Lanczos".  The information that is needed to set up
              the sparse matrices V_dh, R_dh, W_dh, and L_dh is also provided
 	      in the structures "result" returned by these functions.
 
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   BANDITS: a Matlab Package of Band Krylov Subspace Iterations
 
   Copyright (c) 2018-2019 Roland W. Freund
   See LICENSE.txt for license
 
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