| [abd,lda,n,ml,mu,ipvt,rcond,z]=sgbco(abd,lda,n,ml,mu,ipvt,rcond,z); |
function [abd,lda,n,ml,mu,ipvt,rcond,z]=sgbco(abd,lda,n,ml,mu,ipvt,rcond,z);
%***BEGIN PROLOGUE SGBCO
%***PURPOSE Factor a band matrix by Gaussian elimination and
% estimate the condition number of the matrix.
%***LIBRARY SLATEC (LINPACK)
%***CATEGORY D2A2
%***TYPE SINGLE PRECISION (SGBCO-S, DGBCO-D, CGBCO-C)
%***KEYWORDS BANDED, CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
% MATRIX FACTORIZATION
%***AUTHOR Moler, C. B., (U. of New Mexico)
%***DESCRIPTION
%
% SBGCO factors a real band matrix by Gaussian
% elimination and estimates the condition of the matrix.
%
% If RCOND is not needed, SGBFA is slightly faster.
% To solve A*X = B , follow SBGCO by SGBSL.
% To compute INVERSE(A)*C , follow SBGCO by SGBSL.
% To compute DETERMINANT(A) , follow SBGCO by SGBDI.
%
% On Entry
%
% ABD REAL(LDA, N)
% contains the matrix in band storage. The columns
% of the matrix are stored in the columns of ABD and
% the diagonals of the matrix are stored in rows
% ML+1 through 2*ML+MU+1 of ABD .
% See the comments below for details.
%
% LDA INTEGER
% the leading dimension of the array ABD .
% LDA must be .GE. 2*ML + MU + 1 .
%
% N INTEGER
% the order of the original matrix.
%
% ML INTEGER
% number of diagonals below the main diagonal.
% 0 .LE. ML .LT. N .
%
% MU INTEGER
% number of diagonals above the main diagonal.
% 0 .LE. MU .LT. N .
% More efficient if ML .LE. MU .
%
% On Return
%
% ABD an upper triangular matrix in band storage and
% the multipliers which were used to obtain it.
% The factorization can be written A = L*U where
% L is a product of permutation and unit lower
% triangular matrices and U is upper triangular.
%
% IPVT INTEGER(N)
% an integer vector of pivot indices.
%
% RCOND REAL
% an estimate of the reciprocal condition of A .
% For the system A*X = B , relative perturbations
% in A and B of size EPSILON may cause
% relative perturbations in X of size EPSILON/RCOND .
% If RCOND is so small that the logical expression
% 1.0 + RCOND .EQ. 1.0
% is truemlv, then A may be singular to working
% precision. In particular, RCOND is zero if
% exact singularity is detected or the estimate
% underflows.
%
% Z REAL(N)
% a work vector whose contents are usually unimportant.
% If A is close to a singular matrix, then Z is
% an approximate null vector in the sense that
% NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
%
% Band Storage
%
% If A is a band matrix, the following program segment
% will set up the input.
%
% ML = (band width below the diagonal)
% MU = (band width above the diagonal)
% M = ML + MU + 1
% DO 20 J = 1, N
% I1 = MAX(1, J-MU)
% I2 = MIN(N, J+ML)
% DO 10 I = I1, I2
% K = I - J + M
% ABD(K,J) = A(I,J)
% 10 CONTINUE
% 20 CONTINUE
%
% This uses rows ML+1 through 2*ML+MU+1 of ABD .
% In addition, the first ML rows in ABD are used for
% elements generated during the triangularization.
% The total number of rows needed in ABD is 2*ML+MU+1 .
% The ML+MU by ML+MU upper left triangle and the
% ML by ML lower right triangle are not referenced.
%
% Example: If the original matrix is
%
% 11 12 13 0 0 0
% 21 22 23 24 0 0
% 0 32 33 34 35 0
% 0 0 43 44 45 46
% 0 0 0 54 55 56
% 0 0 0 0 65 66
%
% then N = 6, ML = 1, MU = 2, LDA .GE. 5 and ABD should contain
%
% * * * + + + , * = not used
% * * 13 24 35 46 , + = used for pivoting
% * 12 23 34 45 56
% 11 22 33 44 55 66
% 21 32 43 54 65 *
%
%***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
% Stewart, LINPACK Users' Guide, SIAM, 1979.
%***ROUTINES CALLED SASUM, SAXPY, SDOT, SGBFA, SSCAL
%***REVISION HISTORY (YYMMDD)
% 780814 DATE WRITTEN
% 890531 Changed all specific intrinsics to generic. (WRB)
% 890831 Modified array declarations. (WRB)
% 890831 REVISION DATE from Version 3.2
% 891214 Prologue converted to Version 4.0 format. (BAB)
% 900326 Removed duplicate information from DESCRIPTION section.
% (WRB)
% 920501 Reformatted the REFERENCES section. (WRB)
%***end PROLOGUE SGBCO
persistent anorm ek info is j ju k kb kp1 l la lm lz m mm s sm t wk wkm ynorm ;
ipvt_shape=size(ipvt);ipvt=reshape(ipvt,1,[]);
abd_shape=size(abd);abd=reshape([abd(:).',zeros(1,ceil(numel(abd)./prod([lda])).*prod([lda])-numel(abd))],lda,[]);
z_shape=size(z);z=reshape(z,1,[]);
%
if isempty(ek), ek=0; end;
if isempty(t), t=0; end;
if isempty(wk), wk=0; end;
if isempty(wkm), wkm=0; end;
if isempty(anorm), anorm=0; end;
if isempty(s), s=0; end;
if isempty(sm), sm=0; end;
if isempty(ynorm), ynorm=0; end;
if isempty(is), is=0; end;
if isempty(info), info=0; end;
if isempty(j), j=0; end;
if isempty(ju), ju=0; end;
if isempty(k), k=0; end;
if isempty(kb), kb=0; end;
if isempty(kp1), kp1=0; end;
if isempty(l), l=0; end;
if isempty(la), la=0; end;
if isempty(lm), lm=0; end;
if isempty(lz), lz=0; end;
if isempty(m), m=0; end;
if isempty(mm), mm=0; end;
%
% COMPUTE 1-NORM OF A
%
%***FIRST EXECUTABLE STATEMENT SGBCO
anorm = 0.0e0;
l = fix(ml + 1);
is = fix(l + mu);
for j = 1 : n;
anorm = max(anorm,sasum(l,abd(sub2ind(size(abd),is,j):end),1));
if( is>ml+1 )
is = fix(is - 1);
end;
if( j<=mu )
l = fix(l + 1);
end;
if( j>=n-ml )
l = fix(l - 1);
end;
end; j = fix(n+1);
%
% FACTOR
%
[abd,lda,n,ml,mu,ipvt,info]=sgbfa(abd,lda,n,ml,mu,ipvt,info);
%
% RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) .
% ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND TRANS(A)*Y = E .
% TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE
% CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE
% TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID
% OVERFLOW.
%
% SOLVE TRANS(U)*W = E
%
ek = 1.0e0;
for j = 1 : n;
z(j) = 0.0e0;
end; j = fix(n+1);
m = fix(ml + mu + 1);
ju = 0;
for k = 1 : n;
if( z(k)~=0.0e0 )
ek = (abs(ek).*sign(-z(k)));
end;
if( abs(ek-z(k))>abs(abd(m,k)) )
s = abs(abd(m,k))./abs(ek-z(k));
[n,s,z]=sscal(n,s,z,1);
ek = s.*ek;
end;
wk = ek - z(k);
wkm = -ek - z(k);
s = abs(wk);
sm = abs(wkm);
if( abd(m,k)==0.0e0 )
wk = 1.0e0;
wkm = 1.0e0;
else;
wk = wk./abd(m,k);
wkm = wkm./abd(m,k);
end;
kp1 = fix(k + 1);
ju = fix(min(max(ju,mu+ipvt(k)),n));
mm = fix(m);
if( kp1<=ju )
for j = kp1 : ju;
mm = fix(mm - 1);
sm = sm + abs(z(j)+wkm.*abd(mm,j));
z(j) = z(j) + wk.*abd(mm,j);
s = s + abs(z(j));
end; j = fix(ju+1);
if( s<sm )
t = wkm - wk;
wk = wkm;
mm = fix(m);
for j = kp1 : ju;
mm = fix(mm - 1);
z(j) = z(j) + t.*abd(mm,j);
end; j = fix(ju+1);
end;
end;
z(k) = wk;
end; k = fix(n+1);
s = 1.0e0./sasum(n,z,1);
[n,s,z]=sscal(n,s,z,1);
%
% SOLVE TRANS(L)*Y = W
%
for kb = 1 : n;
k = fix(n + 1 - kb);
lm = fix(min(ml,n-k));
if( k<n )
z(k) = z(k) + sdot(lm,abd(sub2ind(size(abd),m+1,k):end),1,z(sub2ind(size(z),max(k+1,1)):end),1);
end;
if( abs(z(k))>1.0e0 )
s = 1.0e0./abs(z(k));
[n,s,z]=sscal(n,s,z,1);
end;
l = fix(ipvt(k));
t = z(l);
z(l) = z(k);
z(k) = t;
end; kb = fix(n+1);
s = 1.0e0./sasum(n,z,1);
[n,s,z]=sscal(n,s,z,1);
%
ynorm = 1.0e0;
%
% SOLVE L*V = Y
%
for k = 1 : n;
l = fix(ipvt(k));
t = z(l);
z(l) = z(k);
z(k) = t;
lm = fix(min(ml,n-k));
if( k<n )
[lm,t,abd(sub2ind(size(abd),m+1,k):end),dumvar4,z(sub2ind(size(z),max(k+1,1)):end)]=saxpy(lm,t,abd(sub2ind(size(abd),m+1,k):end),1,z(sub2ind(size(z),max(k+1,1)):end),1);
end;
if( abs(z(k))>1.0e0 )
s = 1.0e0./abs(z(k));
[n,s,z]=sscal(n,s,z,1);
ynorm = s.*ynorm;
end;
end; k = fix(n+1);
s = 1.0e0./sasum(n,z,1);
[n,s,z]=sscal(n,s,z,1);
ynorm = s.*ynorm;
%
% SOLVE U*Z = W
%
for kb = 1 : n;
k = fix(n + 1 - kb);
if( abs(z(k))>abs(abd(m,k)) )
s = abs(abd(m,k))./abs(z(k));
[n,s,z]=sscal(n,s,z,1);
ynorm = s.*ynorm;
end;
if( abd(m,k)~=0.0e0 )
z(k) = z(k)./abd(m,k);
end;
if( abd(m,k)==0.0e0 )
z(k) = 1.0e0;
end;
lm = fix(min(k,m) - 1);
la = fix(m - lm);
lz = fix(k - lm);
t = -z(k);
[lm,t,abd(sub2ind(size(abd),la,k):end),dumvar4,z(sub2ind(size(z),max(lz,1)):end)]=saxpy(lm,t,abd(sub2ind(size(abd),la,k):end),1,z(sub2ind(size(z),max(lz,1)):end),1);
end; kb = fix(n+1);
% MAKE ZNORM = 1.0
s = 1.0e0./sasum(n,z,1);
[n,s,z]=sscal(n,s,z,1);
ynorm = s.*ynorm;
%
if( anorm~=0.0e0 )
rcond = ynorm./anorm;
end;
if( anorm==0.0e0 )
rcond = 0.0e0;
end;
ipvt_shape=zeros(ipvt_shape);ipvt_shape(:)=ipvt(1:numel(ipvt_shape));ipvt=ipvt_shape;
abd_shape=zeros(abd_shape);abd_shape(:)=abd(1:numel(abd_shape));abd=abd_shape;
z_shape=zeros(z_shape);z_shape(:)=z(1:numel(z_shape));z=z_shape;
end
%DECK SGBDI
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