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// Copyright Bruno Pinçon, ESIAL-IECN, Inria CORIDA project
// <bruno.pincon@iecn.u-nancy.fr>
//
// This set of scilab 's macros provide a few sparse utilities.
//
// This software is governed by the CeCILL license under French law and
// abiding by the rules of distribution of free software. You can use,
// modify and/ or redistribute the software under the terms of the CeCILL
// license as circulated by CEA, CNRS and INRIA at the following URL
// "http://www.cecill.info".
//
// As a counterpart to the access to the source code and rights to copy,
// modify and redistribute granted by the license, users are provided only
// with a limited warranty and the software's author, the holder of the
// economic rights, and the successive licensors have only limited
// liability.
//
// In this respect, the user's attention is drawn to the risks associated
// with loading, using, modifying and/or developing or reproducing the
// software by the user in light of its specific status of free software,
// that may mean that it is complicated to manipulate, and that also
// therefore means that it is reserved for developers and experienced
// professionals having in-depth computer knowledge. Users are therefore
// encouraged to load and test the software's suitability as regards their
// requirements in conditions enabling the security of their systems and/or
// data to be ensured and, more generally, to use and operate it in the
// same conditions as regards security.
//
// The fact that you are presently reading this means that you have had
// knowledge of the CeCILL license and that you accept its terms.
function [K1] = condestsp(A, arg2, arg3)
// (A, LUp , t)
// PURPOSE
// Give an estimate of the 1-norm condition number of
// the sparse matrix A by Algorithm 2.4 appearing in :
//
// "A block algorithm for matrix 1-norm estimation
// with an application to 1-norm pseudospectra"
// Nicholas J. Higham and Francoise Tisseur
// Siam J. Matrix Anal. Appl., vol 21, No 4, pp 1185-1201
//
// PARAMETERS
// A : a square sparse matrix
//
// LUp : (optional) a pointer to (umf) LU factors of A
// gotten by a call to umf_lufact ; if you
// have already computed the LU (= PAQ) factors
// it is recommanded to give this optional
// parameter (as the factorization may be time
// consuming)
//
// t : (optional) a positive integer
//
// K1 : estimated 1-norm condition number of A
//
// POSSIBLE CALLING SEQUENCES
// [K1, [x]] = condestsp(A, LUp, t)
// [K1, [x]] = condestsp(A, LUp)
// [K1, [x]] = condestsp(A, t)
// [K1, [x]] = condestsp(A)
//
// AUTHOR
// Bruno Pincon <Bruno.Pincon@iecn.u-nancy.fr> but nearly
// close to the given algorithm as this one is written in
// a "matlab-like" language
//
[lhs, rhs] = argn()
if rhs<1 | rhs>3 | lhs > 2 then
error(msprintf(gettext("%s: Wrong number of input arguments: %d to %d expected.\n"),"condestsp",1,3))
end
if typeof(A) ~= "sparse" then
error(msprintf(gettext("%s: Wrong type for input argument #%d: Square sparse matrix expected.\n"),"condestsp",1))
else
[n,m] = size(A)
if n~=m then
error(msprintf(gettext("%s: Wrong size for input argument #%d: Square sparse matrix expected.\n"),"condestsp",1))
end
end
factor_inside = %f // when LUp is given (after the following tests
// this var is set to %t if the factorisation
// is computed inside this function)
if rhs == 1 then
LUp = umf_lufact(A) ; factor_inside = %t ; t = 2
elseif rhs == 2 then
if typeof(arg2) == "pointer" then
LUp = arg2 ; t = 2
else
t = arg2 ; LUp = umf_lufact(A) ; factor_inside = %t
end
elseif rhs == 3 then
LUp = arg2 ; t = arg3
end
// verify if LUp and T are valid !
[OK, nrow, ncol] = umf_luinfo(LUp)
if ~OK then
error(" the LU pointer is not valid")
elseif n ~= nrow | nrow ~= ncol
error(msprintf(gettext("%s: The matrix and the LU factors have not the same dimension !\n"),"condestsp"));
end
if int(t)~=t | length(t)~=1 | or(t < 1) then
error(msprintf(gettext("%s: Invalid type and/or size and/or value for the second arg.\n"),"condestsp"));
end
// go on
// the algo need a fortran-like sign function (with sign(0) = 1
// and not with sign(0)=0 as the scilab native 's one)
deff("s = fsign(x)", "s = sign(x) ; s(find(s == 0)) = 1")
// Part 1 : computes ||A||_1
norm1_A = norm(A,1)
// Part 2 : computes (estimates) || A^(-1) ||_1
// 1/ choose starting matrix X (n,t)
X = ones(n,t)
X(:,2:t) = fsign(rand(n,t-1)-0.5)
X = test_on_columns(X) / n
Y = zeros(X) ; Z = zeros(X)
ind_hist = []
est_old = 0
ind = zeros(n,1)
S = zeros(n,t)
k = 1 ; itmax = 5
while %t
// solve Y = A^(-1) X <=> A Y = X
for j=1:t
Y(:,j) = umf_lusolve(LUp, X(:,j))
end
[est, ind_est] = max( sum(abs(Y),"r") )
if est > est_old | k==2 then
ind_best = ind_est
w = Y(:,ind_best)
end
if k >= 2 & est <= est_old then, est = est_old, break, end
est_old = est ; S_old = S
if k > itmax then , break , end
S = fsign(Y)
// if every column of S is // to a column of S_old then it is finish
if and( abs(S_old'*S) == n ) then , break , end
if t > 1 then
// s'assurer qu'aucune colonne de S n'est // a une autre
// ou a une colonne de S_old en remplacant des colonnes par rand{-1,1}
S = test_on_columns(S,S_old)
end
// calcul de Z = A' S
for j=1:t
Z(:,j) = umf_lusolve(LUp, S(:,j),"A''x=b")
end
[h,ind] = gsort(max(abs(Z),"c"))
if k >= 2 then
if h(1) == h(ind_best) then , break , end
end
if (t > 1) & (k > 1) then
j = 1
for l=1:t
while %t & (j<=length(ind))
if find(ind_hist == ind(j)) == [] then
ind(l) = ind(j)
j = j + 1
break
else
j = j + 1
end
end
end
end
X = zeros(n,t)
for l = 1:t
X(ind(l),l) = 1
end
ind_hist = [ind_hist ; ind(1:t)]
k = k + 1
end
K1 = est * norm1_A
if factor_inside then
umf_ludel(LUp)
end
endfunction
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