static void testrotations()
{
ap::real_2d_array al1;
ap::real_2d_array al2;
ap::real_2d_array ar1;
ap::real_2d_array ar2;
ap::real_1d_array cl;
ap::real_1d_array sl;
ap::real_1d_array cr;
ap::real_1d_array sr;
ap::real_1d_array w;
int m;
int n;
int maxmn;
double t;
int pass;
int passcount;
int i;
int j;
double err;
double maxerr;
bool isforward;
passcount = 1000;
maxerr = 0;
for(pass = 1; pass <= passcount; pass++)
{
//
// settings
//
m = 2+ap::randominteger(50);
n = 2+ap::randominteger(50);
isforward = ap::randomreal()>0.5;
maxmn = ap::maxint(m, n);
al1.setbounds(1, m, 1, n);
al2.setbounds(1, m, 1, n);
ar1.setbounds(1, m, 1, n);
ar2.setbounds(1, m, 1, n);
cl.setbounds(1, m-1);
sl.setbounds(1, m-1);
cr.setbounds(1, n-1);
sr.setbounds(1, n-1);
w.setbounds(1, maxmn);
//
// matrices and rotaions
//
for(i = 1; i <= m; i++)
{
for(j = 1; j <= n; j++)
{
al1(i,j) = 2*ap::randomreal()-1;
al2(i,j) = al1(i,j);
ar1(i,j) = al1(i,j);
ar2(i,j) = al1(i,j);
}
}
for(i = 1; i <= m-1; i++)
{
t = 2*ap::pi()*ap::randomreal();
cl(i) = cos(t);
sl(i) = sin(t);
}
for(j = 1; j <= n-1; j++)
{
t = 2*ap::pi()*ap::randomreal();
cr(j) = cos(t);
sr(j) = sin(t);
}
//
// Test left
//
applyrotationsfromtheleft(isforward, 1, m, 1, n, cl, sl, al1, w);
for(j = 1; j <= n; j++)
{
applyrotationsfromtheleft(isforward, 1, m, j, j, cl, sl, al2, w);
}
err = 0;
for(i = 1; i <= m; i++)
{
for(j = 1; j <= n; j++)
{
err = ap::maxreal(err, fabs(al1(i,j)-al2(i,j)));
}
}
maxerr = ap::maxreal(err, maxerr);
//
// Test right
//
applyrotationsfromtheright(isforward, 1, m, 1, n, cr, sr, ar1, w);
for(i = 1; i <= m; i++)
{
applyrotationsfromtheright(isforward, i, i, 1, n, cr, sr, ar2, w);
}
err = 0;
for(i = 1; i <= m; i++)
{
for(j = 1; j <= n; j++)
{
err = ap::maxreal(err, fabs(ar1(i,j)-ar2(i,j)));
}
}
maxerr = ap::maxreal(err, maxerr);
}
printf("TESTING ROTATIONS\n");
printf("Pass count %0ld\n",
long(passcount));
printf("Error is %5.3le\n",
double(maxerr));
}
static void internalauxschur(bool wantt,
bool wantz,
int n,
int ilo,
int ihi,
ap::real_2d_array& h,
ap::real_1d_array& wr,
ap::real_1d_array& wi,
int iloz,
int ihiz,
ap::real_2d_array& z,
ap::real_1d_array& work,
ap::real_1d_array& workv3,
ap::real_1d_array& workc1,
ap::real_1d_array& works1,
int& info)
{
int i;
int i1;
int i2;
int itn;
int its;
int j;
int k;
int l;
int m;
int nh;
int nr;
int nz;
double ave;
double cs;
double disc;
double h00;
double h10;
double h11;
double h12;
double h21;
double h22;
double h33;
double h33s;
double h43h34;
double h44;
double h44s;
double ovfl;
double s;
double smlnum;
double sn;
double sum;
double t1;
double t2;
double t3;
double tst1;
double unfl;
double v1;
double v2;
double v3;
bool failflag;
double dat1;
double dat2;
int p1;
double him1im1;
double him1i;
double hiim1;
double hii;
double wrim1;
double wri;
double wiim1;
double wii;
double ulp;
info = 0;
dat1 = 0.75;
dat2 = -0.4375;
ulp = ap::machineepsilon;
//
// Quick return if possible
//
if( n==0 )
{
return;
}
if( ilo==ihi )
{
wr(ilo) = h(ilo,ilo);
wi(ilo) = 0;
return;
}
nh = ihi-ilo+1;
nz = ihiz-iloz+1;
//
// Set machine-dependent constants for the stopping criterion.
// If norm(H) <= sqrt(OVFL), overflow should not occur.
//
unfl = ap::minrealnumber;
ovfl = 1/unfl;
smlnum = unfl*(nh/ulp);
//
// I1 and I2 are the indices of the first row and last column of H
// to which transformations must be applied. If eigenvalues only are
// being computed, I1 and I2 are set inside the main loop.
//
if( wantt )
{
i1 = 1;
i2 = n;
}
//
// ITN is the total number of QR iterations allowed.
//
itn = 30*nh;
//
// The main loop begins here. I is the loop index and decreases from
// IHI to ILO in steps of 1 or 2. Each iteration of the loop works
// with the active submatrix in rows and columns L to I.
// Eigenvalues I+1 to IHI have already converged. Either L = ILO or
// H(L,L-1) is negligible so that the matrix splits.
//
i = ihi;
while(true)
{
l = ilo;
if( i<ilo )
{
return;
}
//
// Perform QR iterations on rows and columns ILO to I until a
// submatrix of order 1 or 2 splits off at the bottom because a
// subdiagonal element has become negligible.
//
failflag = true;
for(its = 0; its <= itn; its++)
{
//
// Look for a single small subdiagonal element.
//
for(k = i; k >= l+1; k--)
{
tst1 = fabs(h(k-1,k-1))+fabs(h(k,k));
if( tst1==0 )
{
tst1 = upperhessenberg1norm(h, l, i, l, i, work);
}
if( fabs(h(k,k-1))<=ap::maxreal(ulp*tst1, smlnum) )
{
break;
}
}
l = k;
if( l>ilo )
{
//
// H(L,L-1) is negligible
//
h(l,l-1) = 0;
}
//
// Exit from loop if a submatrix of order 1 or 2 has split off.
//
if( l>=i-1 )
{
failflag = false;
break;
}
//
// Now the active submatrix is in rows and columns L to I. If
// eigenvalues only are being computed, only the active submatrix
// need be transformed.
//
if( !wantt )
{
i1 = l;
i2 = i;
}
if( its==10||its==20 )
{
//
// Exceptional shift.
//
s = fabs(h(i,i-1))+fabs(h(i-1,i-2));
h44 = dat1*s+h(i,i);
h33 = h44;
h43h34 = dat2*s*s;
}
else
{
//
// Prepare to use Francis' double shift
// (i.e. 2nd degree generalized Rayleigh quotient)
//
h44 = h(i,i);
h33 = h(i-1,i-1);
h43h34 = h(i,i-1)*h(i-1,i);
s = h(i-1,i-2)*h(i-1,i-2);
disc = (h33-h44)*0.5;
disc = disc*disc+h43h34;
if( disc>0 )
{
//
// Real roots: use Wilkinson's shift twice
//
disc = sqrt(disc);
ave = 0.5*(h33+h44);
if( fabs(h33)-fabs(h44)>0 )
{
h33 = h33*h44-h43h34;
h44 = h33/(extschursign(disc, ave)+ave);
}
else
{
h44 = extschursign(disc, ave)+ave;
}
h33 = h44;
h43h34 = 0;
}
}
//
// Look for two consecutive small subdiagonal elements.
//
for(m = i-2; m >= l; m--)
{
//
// Determine the effect of starting the double-shift QR
// iteration at row M, and see if this would make H(M,M-1)
// negligible.
//
h11 = h(m,m);
h22 = h(m+1,m+1);
h21 = h(m+1,m);
h12 = h(m,m+1);
h44s = h44-h11;
h33s = h33-h11;
v1 = (h33s*h44s-h43h34)/h21+h12;
v2 = h22-h11-h33s-h44s;
v3 = h(m+2,m+1);
s = fabs(v1)+fabs(v2)+fabs(v3);
v1 = v1/s;
v2 = v2/s;
v3 = v3/s;
workv3(1) = v1;
workv3(2) = v2;
workv3(3) = v3;
if( m==l )
{
break;
}
h00 = h(m-1,m-1);
h10 = h(m,m-1);
tst1 = fabs(v1)*(fabs(h00)+fabs(h11)+fabs(h22));
if( fabs(h10)*(fabs(v2)+fabs(v3))<=ulp*tst1 )
{
break;
}
}
//
// Double-shift QR step
//
for(k = m; k <= i-1; k++)
{
//
// The first iteration of this loop determines a reflection G
// from the vector V and applies it from left and right to H,
// thus creating a nonzero bulge below the subdiagonal.
//
// Each subsequent iteration determines a reflection G to
// restore the Hessenberg form in the (K-1)th column, and thus
// chases the bulge one step toward the bottom of the active
// submatrix. NR is the order of G.
//
nr = ap::minint(3, i-k+1);
if( k>m )
{
for(p1 = 1; p1 <= nr; p1++)
{
workv3(p1) = h(k+p1-1,k-1);
}
}
generatereflection(workv3, nr, t1);
if( k>m )
{
h(k,k-1) = workv3(1);
h(k+1,k-1) = 0;
if( k<i-1 )
{
h(k+2,k-1) = 0;
}
}
else
{
if( m>l )
{
h(k,k-1) = -h(k,k-1);
}
}
v2 = workv3(2);
t2 = t1*v2;
if( nr==3 )
{
v3 = workv3(3);
t3 = t1*v3;
//
// Apply G from the left to transform the rows of the matrix
// in columns K to I2.
//
for(j = k; j <= i2; j++)
{
sum = h(k,j)+v2*h(k+1,j)+v3*h(k+2,j);
h(k,j) = h(k,j)-sum*t1;
h(k+1,j) = h(k+1,j)-sum*t2;
h(k+2,j) = h(k+2,j)-sum*t3;
}
//
// Apply G from the right to transform the columns of the
// matrix in rows I1 to min(K+3,I).
//
for(j = i1; j <= ap::minint(k+3, i); j++)
{
sum = h(j,k)+v2*h(j,k+1)+v3*h(j,k+2);
h(j,k) = h(j,k)-sum*t1;
h(j,k+1) = h(j,k+1)-sum*t2;
h(j,k+2) = h(j,k+2)-sum*t3;
}
if( wantz )
{
//
// Accumulate transformations in the matrix Z
//
for(j = iloz; j <= ihiz; j++)
{
sum = z(j,k)+v2*z(j,k+1)+v3*z(j,k+2);
z(j,k) = z(j,k)-sum*t1;
z(j,k+1) = z(j,k+1)-sum*t2;
z(j,k+2) = z(j,k+2)-sum*t3;
}
}
}
else
{
if( nr==2 )
{
//
// Apply G from the left to transform the rows of the matrix
// in columns K to I2.
//
for(j = k; j <= i2; j++)
{
sum = h(k,j)+v2*h(k+1,j);
h(k,j) = h(k,j)-sum*t1;
h(k+1,j) = h(k+1,j)-sum*t2;
}
//
// Apply G from the right to transform the columns of the
// matrix in rows I1 to min(K+3,I).
//
for(j = i1; j <= i; j++)
{
sum = h(j,k)+v2*h(j,k+1);
h(j,k) = h(j,k)-sum*t1;
h(j,k+1) = h(j,k+1)-sum*t2;
}
if( wantz )
{
//
// Accumulate transformations in the matrix Z
//
for(j = iloz; j <= ihiz; j++)
{
sum = z(j,k)+v2*z(j,k+1);
z(j,k) = z(j,k)-sum*t1;
z(j,k+1) = z(j,k+1)-sum*t2;
}
}
}
}
}
}
if( failflag )
{
//
// Failure to converge in remaining number of iterations
//
info = i;
return;
}
if( l==i )
{
//
// H(I,I-1) is negligible: one eigenvalue has converged.
//
wr(i) = h(i,i);
wi(i) = 0;
}
else
{
if( l==i-1 )
{
//
// H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
//
// Transform the 2-by-2 submatrix to standard Schur form,
// and compute and store the eigenvalues.
//
him1im1 = h(i-1,i-1);
him1i = h(i-1,i);
hiim1 = h(i,i-1);
hii = h(i,i);
aux2x2schur(him1im1, him1i, hiim1, hii, wrim1, wiim1, wri, wii, cs, sn);
wr(i-1) = wrim1;
wi(i-1) = wiim1;
wr(i) = wri;
wi(i) = wii;
h(i-1,i-1) = him1im1;
h(i-1,i) = him1i;
h(i,i-1) = hiim1;
h(i,i) = hii;
if( wantt )
{
//
// Apply the transformation to the rest of H.
//
if( i2>i )
{
workc1(1) = cs;
works1(1) = sn;
applyrotationsfromtheleft(true, i-1, i, i+1, i2, workc1, works1, h, work);
}
workc1(1) = cs;
works1(1) = sn;
applyrotationsfromtheright(true, i1, i-2, i-1, i, workc1, works1, h, work);
}
if( wantz )
{
//
// Apply the transformation to Z.
//
workc1(1) = cs;
works1(1) = sn;
applyrotationsfromtheright(true, iloz, iloz+nz-1, i-1, i, workc1, works1, z, work);
}
}
}
//
// Decrement number of remaining iterations, and return to start of
// the main loop with new value of I.
//
itn = itn-its;
i = l-1;
}
}
/*************************************************************************
Internal working subroutine for bidiagonal decomposition
*************************************************************************/
bool bidiagonalsvddecompositioninternal(ap::real_1d_array& d,
ap::real_1d_array e,
int n,
bool isupper,
bool isfractionalaccuracyrequired,
ap::real_2d_array& u,
int ustart,
int nru,
ap::real_2d_array& c,
int cstart,
int ncc,
ap::real_2d_array& vt,
int vstart,
int ncvt)
{
bool result;
int i;
int idir;
int isub;
int iter;
int j;
int ll = 0; // Eliminate compiler warning.
int lll;
int m;
int maxit;
int oldll;
int oldm;
double abse;
double abss;
double cosl;
double cosr;
double cs;
double eps;
double f;
double g;
double h;
double mu;
double oldcs;
double oldsn = 0.; // Eliminate compiler warning.
double r;
double shift;
double sigmn;
double sigmx;
double sinl;
double sinr;
double sll;
double smax;
double smin;
double sminl;
double sminoa;
double sn;
double thresh;
double tol;
double tolmul;
double unfl;
ap::real_1d_array work0;
ap::real_1d_array work1;
ap::real_1d_array work2;
ap::real_1d_array work3;
int maxitr;
bool matrixsplitflag;
bool iterflag;
ap::real_1d_array utemp;
ap::real_1d_array vttemp;
ap::real_1d_array ctemp;
ap::real_1d_array etemp;
bool fwddir;
double tmp;
int mm1;
int mm0;
bool bchangedir;
int uend;
int cend;
int vend;
result = true;
if( n==0 )
{
return result;
}
if( n==1 )
{
if( d(1)<0 )
{
d(1) = -d(1);
if( ncvt>0 )
{
ap::vmul(&vt(vstart, vstart), ap::vlen(vstart,vstart+ncvt-1), -1);
}
}
return result;
}
//
// init
//
work0.setbounds(1, n-1);
work1.setbounds(1, n-1);
work2.setbounds(1, n-1);
work3.setbounds(1, n-1);
uend = ustart+ap::maxint(nru-1, 0);
vend = vstart+ap::maxint(ncvt-1, 0);
cend = cstart+ap::maxint(ncc-1, 0);
utemp.setbounds(ustart, uend);
vttemp.setbounds(vstart, vend);
ctemp.setbounds(cstart, cend);
maxitr = 12;
fwddir = true;
//
// resize E from N-1 to N
//
etemp.setbounds(1, n);
for(i = 1; i <= n-1; i++)
{
etemp(i) = e(i);
}
e.setbounds(1, n);
for(i = 1; i <= n-1; i++)
{
e(i) = etemp(i);
}
e(n) = 0;
idir = 0;
//
// Get machine constants
//
eps = ap::machineepsilon;
unfl = ap::minrealnumber;
//
// If matrix lower bidiagonal, rotate to be upper bidiagonal
// by applying Givens rotations on the left
//
if( !isupper )
{
for(i = 1; i <= n-1; i++)
{
generaterotation(d(i), e(i), cs, sn, r);
d(i) = r;
e(i) = sn*d(i+1);
d(i+1) = cs*d(i+1);
work0(i) = cs;
work1(i) = sn;
}
//
// Update singular vectors if desired
//
if( nru>0 )
{
applyrotationsfromtheright(fwddir, ustart, uend, 1+ustart-1, n+ustart-1, work0, work1, u, utemp);
}
if( ncc>0 )
{
applyrotationsfromtheleft(fwddir, 1+cstart-1, n+cstart-1, cstart, cend, work0, work1, c, ctemp);
}
}
//
// Compute singular values to relative accuracy TOL
// (By setting TOL to be negative, algorithm will compute
// singular values to absolute accuracy ABS(TOL)*norm(input matrix))
//
tolmul = ap::maxreal(double(10), ap::minreal(double(100), pow(eps, -0.125)));
tol = tolmul*eps;
if( !isfractionalaccuracyrequired )
{
tol = -tol;
}
//
// Compute approximate maximum, minimum singular values
//
smax = 0;
for(i = 1; i <= n; i++)
{
smax = ap::maxreal(smax, fabs(d(i)));
}
for(i = 1; i <= n-1; i++)
{
smax = ap::maxreal(smax, fabs(e(i)));
}
sminl = 0;
if( tol>=0 )
{
//
// Relative accuracy desired
//
sminoa = fabs(d(1));
if( sminoa!=0 )
{
mu = sminoa;
for(i = 2; i <= n; i++)
{
mu = fabs(d(i))*(mu/(mu+fabs(e(i-1))));
sminoa = ap::minreal(sminoa, mu);
if( sminoa==0 )
{
break;
}
}
}
sminoa = sminoa/sqrt(double(n));
thresh = ap::maxreal(tol*sminoa, maxitr*n*n*unfl);
}
else
{
//
// Absolute accuracy desired
//
thresh = ap::maxreal(fabs(tol)*smax, maxitr*n*n*unfl);
}
//
// Prepare for main iteration loop for the singular values
// (MAXIT is the maximum number of passes through the inner
// loop permitted before nonconvergence signalled.)
//
maxit = maxitr*n*n;
iter = 0;
oldll = -1;
oldm = -1;
//
// M points to last element of unconverged part of matrix
//
m = n;
//
// Begin main iteration loop
//
while(true)
{
//
// Check for convergence or exceeding iteration count
//
if( m<=1 )
{
break;
}
if( iter>maxit )
{
result = false;
return result;
}
//
// Find diagonal block of matrix to work on
//
if( tol<0&&fabs(d(m))<=thresh )
{
d(m) = 0;
}
smax = fabs(d(m));
smin = smax;
matrixsplitflag = false;
for(lll = 1; lll <= m-1; lll++)
{
ll = m-lll;
abss = fabs(d(ll));
abse = fabs(e(ll));
if( tol<0&&abss<=thresh )
{
d(ll) = 0;
}
if( abse<=thresh )
{
matrixsplitflag = true;
break;
}
smin = ap::minreal(smin, abss);
smax = ap::maxreal(smax, ap::maxreal(abss, abse));
}
if( !matrixsplitflag )
{
ll = 0;
}
else
{
//
// Matrix splits since E(LL) = 0
//
e(ll) = 0;
if( ll==m-1 )
{
//
// Convergence of bottom singular value, return to top of loop
//
m = m-1;
continue;
}
}
ll = ll+1;
//
// E(LL) through E(M-1) are nonzero, E(LL-1) is zero
//
if( ll==m-1 )
{
//
// 2 by 2 block, handle separately
//
svdv2x2(d(m-1), e(m-1), d(m), sigmn, sigmx, sinr, cosr, sinl, cosl);
d(m-1) = sigmx;
e(m-1) = 0;
d(m) = sigmn;
//
// Compute singular vectors, if desired
//
if( ncvt>0 )
{
mm0 = m+(vstart-1);
mm1 = m-1+(vstart-1);
ap::vmove(&vttemp(vstart), &vt(mm1, vstart), ap::vlen(vstart,vend), cosr);
ap::vadd(&vttemp(vstart), &vt(mm0, vstart), ap::vlen(vstart,vend), sinr);
ap::vmul(&vt(mm0, vstart), ap::vlen(vstart,vend), cosr);
ap::vsub(&vt(mm0, vstart), &vt(mm1, vstart), ap::vlen(vstart,vend), sinr);
ap::vmove(&vt(mm1, vstart), &vttemp(vstart), ap::vlen(vstart,vend));
}
if( nru>0 )
{
mm0 = m+ustart-1;
mm1 = m-1+ustart-1;
ap::vmove(utemp.getvector(ustart, uend), u.getcolumn(mm1, ustart, uend), cosl);
ap::vadd(utemp.getvector(ustart, uend), u.getcolumn(mm0, ustart, uend), sinl);
ap::vmul(u.getcolumn(mm0, ustart, uend), cosl);
ap::vsub(u.getcolumn(mm0, ustart, uend), u.getcolumn(mm1, ustart, uend), sinl);
ap::vmove(u.getcolumn(mm1, ustart, uend), utemp.getvector(ustart, uend));
}
if( ncc>0 )
{
mm0 = m+cstart-1;
mm1 = m-1+cstart-1;
ap::vmove(&ctemp(cstart), &c(mm1, cstart), ap::vlen(cstart,cend), cosl);
ap::vadd(&ctemp(cstart), &c(mm0, cstart), ap::vlen(cstart,cend), sinl);
ap::vmul(&c(mm0, cstart), ap::vlen(cstart,cend), cosl);
ap::vsub(&c(mm0, cstart), &c(mm1, cstart), ap::vlen(cstart,cend), sinl);
ap::vmove(&c(mm1, cstart), &ctemp(cstart), ap::vlen(cstart,cend));
}
m = m-2;
continue;
}
//
// If working on new submatrix, choose shift direction
// (from larger end diagonal element towards smaller)
//
// Previously was
// "if (LL>OLDM) or (M<OLDLL) then"
// fixed thanks to Michael Rolle < *****@*****.** >
// Very strange that LAPACK still contains it.
//
bchangedir = false;
if( idir==1&&fabs(d(ll))<1.0E-3*fabs(d(m)) )
{
bchangedir = true;
}
if( idir==2&&fabs(d(m))<1.0E-3*fabs(d(ll)) )
{
bchangedir = true;
}
if( ll!=oldll||m!=oldm||bchangedir )
{
if( fabs(d(ll))>=fabs(d(m)) )
{
//
// Chase bulge from top (big end) to bottom (small end)
//
idir = 1;
}
else
{
//
// Chase bulge from bottom (big end) to top (small end)
//
idir = 2;
}
}
//
// Apply convergence tests
//
if( idir==1 )
{
//
// Run convergence test in forward direction
// First apply standard test to bottom of matrix
//
if( (fabs(e(m-1))<=fabs(tol)*fabs(d(m)))||(tol<0&&fabs(e(m-1))<=thresh) )
{
e(m-1) = 0;
continue;
}
if( tol>=0 )
{
//
// If relative accuracy desired,
// apply convergence criterion forward
//
mu = fabs(d(ll));
sminl = mu;
iterflag = false;
for(lll = ll; lll <= m-1; lll++)
{
if( fabs(e(lll))<=tol*mu )
{
e(lll) = 0;
iterflag = true;
break;
}
mu = fabs(d(lll+1))*(mu/(mu+fabs(e(lll))));
sminl = ap::minreal(sminl, mu);
}
if( iterflag )
{
continue;
}
}
}
else
{
//
// Run convergence test in backward direction
// First apply standard test to top of matrix
//
if( (fabs(e(ll))<=fabs(tol)*fabs(d(ll)))||(tol<0&&fabs(e(ll))<=thresh) )
{
e(ll) = 0;
continue;
}
if( tol>=0 )
{
//
// If relative accuracy desired,
// apply convergence criterion backward
//
mu = fabs(d(m));
sminl = mu;
iterflag = false;
for(lll = m-1; lll >= ll; lll--)
{
if( fabs(e(lll))<=tol*mu )
{
e(lll) = 0;
iterflag = true;
break;
}
mu = fabs(d(lll))*(mu/(mu+fabs(e(lll))));
sminl = ap::minreal(sminl, mu);
}
if( iterflag )
{
continue;
}
}
}
oldll = ll;
oldm = m;
//
// Compute shift. First, test if shifting would ruin relative
// accuracy, and if so set the shift to zero.
//
if( tol>=0&&n*tol*(sminl/smax)<=ap::maxreal(eps, 0.01*tol) )
{
//
// Use a zero shift to avoid loss of relative accuracy
//
shift = 0;
}
else
{
//
// Compute the shift from 2-by-2 block at end of matrix
//
if( idir==1 )
{
sll = fabs(d(ll));
svd2x2(d(m-1), e(m-1), d(m), shift, r);
}
else
{
sll = fabs(d(m));
svd2x2(d(ll), e(ll), d(ll+1), shift, r);
}
//
// Test if shift negligible, and if so set to zero
//
if( sll>0 )
{
if( ap::sqr(shift/sll)<eps )
{
shift = 0;
}
}
}
//
// Increment iteration count
//
iter = iter+m-ll;
//
// If SHIFT = 0, do simplified QR iteration
//
if( shift==0 )
{
if( idir==1 )
{
//
// Chase bulge from top to bottom
// Save cosines and sines for later singular vector updates
//
cs = 1;
oldcs = 1;
for(i = ll; i <= m-1; i++)
{
generaterotation(d(i)*cs, e(i), cs, sn, r);
if( i>ll )
{
e(i-1) = oldsn*r;
}
generaterotation(oldcs*r, d(i+1)*sn, oldcs, oldsn, tmp);
d(i) = tmp;
work0(i-ll+1) = cs;
work1(i-ll+1) = sn;
work2(i-ll+1) = oldcs;
work3(i-ll+1) = oldsn;
}
h = d(m)*cs;
d(m) = h*oldcs;
e(m-1) = h*oldsn;
//
// Update singular vectors
//
if( ncvt>0 )
{
applyrotationsfromtheleft(fwddir, ll+vstart-1, m+vstart-1, vstart, vend, work0, work1, vt, vttemp);
}
if( nru>0 )
{
applyrotationsfromtheright(fwddir, ustart, uend, ll+ustart-1, m+ustart-1, work2, work3, u, utemp);
}
if( ncc>0 )
{
applyrotationsfromtheleft(fwddir, ll+cstart-1, m+cstart-1, cstart, cend, work2, work3, c, ctemp);
}
//
// Test convergence
//
if( fabs(e(m-1))<=thresh )
{
e(m-1) = 0;
}
}
else
{
//
// Chase bulge from bottom to top
// Save cosines and sines for later singular vector updates
//
cs = 1;
oldcs = 1;
for(i = m; i >= ll+1; i--)
{
generaterotation(d(i)*cs, e(i-1), cs, sn, r);
if( i<m )
{
e(i) = oldsn*r;
}
generaterotation(oldcs*r, d(i-1)*sn, oldcs, oldsn, tmp);
d(i) = tmp;
work0(i-ll) = cs;
work1(i-ll) = -sn;
work2(i-ll) = oldcs;
work3(i-ll) = -oldsn;
}
h = d(ll)*cs;
d(ll) = h*oldcs;
e(ll) = h*oldsn;
//
// Update singular vectors
//
if( ncvt>0 )
{
applyrotationsfromtheleft(!fwddir, ll+vstart-1, m+vstart-1, vstart, vend, work2, work3, vt, vttemp);
}
if( nru>0 )
{
applyrotationsfromtheright(!fwddir, ustart, uend, ll+ustart-1, m+ustart-1, work0, work1, u, utemp);
}
if( ncc>0 )
{
applyrotationsfromtheleft(!fwddir, ll+cstart-1, m+cstart-1, cstart, cend, work0, work1, c, ctemp);
}
//
// Test convergence
//
if( fabs(e(ll))<=thresh )
{
e(ll) = 0;
}
}
}
else
{
//
// Use nonzero shift
//
if( idir==1 )
{
//
// Chase bulge from top to bottom
// Save cosines and sines for later singular vector updates
//
f = (fabs(d(ll))-shift)*(extsignbdsqr(double(1), d(ll))+shift/d(ll));
g = e(ll);
for(i = ll; i <= m-1; i++)
{
generaterotation(f, g, cosr, sinr, r);
if( i>ll )
{
e(i-1) = r;
}
f = cosr*d(i)+sinr*e(i);
e(i) = cosr*e(i)-sinr*d(i);
g = sinr*d(i+1);
d(i+1) = cosr*d(i+1);
generaterotation(f, g, cosl, sinl, r);
d(i) = r;
f = cosl*e(i)+sinl*d(i+1);
d(i+1) = cosl*d(i+1)-sinl*e(i);
if( i<m-1 )
{
g = sinl*e(i+1);
e(i+1) = cosl*e(i+1);
}
work0(i-ll+1) = cosr;
work1(i-ll+1) = sinr;
work2(i-ll+1) = cosl;
work3(i-ll+1) = sinl;
}
e(m-1) = f;
//
// Update singular vectors
//
if( ncvt>0 )
{
applyrotationsfromtheleft(fwddir, ll+vstart-1, m+vstart-1, vstart, vend, work0, work1, vt, vttemp);
}
if( nru>0 )
{
applyrotationsfromtheright(fwddir, ustart, uend, ll+ustart-1, m+ustart-1, work2, work3, u, utemp);
}
if( ncc>0 )
{
applyrotationsfromtheleft(fwddir, ll+cstart-1, m+cstart-1, cstart, cend, work2, work3, c, ctemp);
}
//
// Test convergence
//
if( fabs(e(m-1))<=thresh )
{
e(m-1) = 0;
}
}
else
{
//
// Chase bulge from bottom to top
// Save cosines and sines for later singular vector updates
//
f = (fabs(d(m))-shift)*(extsignbdsqr(double(1), d(m))+shift/d(m));
g = e(m-1);
for(i = m; i >= ll+1; i--)
{
generaterotation(f, g, cosr, sinr, r);
if( i<m )
{
e(i) = r;
}
f = cosr*d(i)+sinr*e(i-1);
e(i-1) = cosr*e(i-1)-sinr*d(i);
g = sinr*d(i-1);
d(i-1) = cosr*d(i-1);
generaterotation(f, g, cosl, sinl, r);
d(i) = r;
f = cosl*e(i-1)+sinl*d(i-1);
d(i-1) = cosl*d(i-1)-sinl*e(i-1);
if( i>ll+1 )
{
g = sinl*e(i-2);
e(i-2) = cosl*e(i-2);
}
work0(i-ll) = cosr;
work1(i-ll) = -sinr;
work2(i-ll) = cosl;
work3(i-ll) = -sinl;
}
e(ll) = f;
//
// Test convergence
//
if( fabs(e(ll))<=thresh )
{
e(ll) = 0;
}
//
// Update singular vectors if desired
//
if( ncvt>0 )
{
applyrotationsfromtheleft(!fwddir, ll+vstart-1, m+vstart-1, vstart, vend, work2, work3, vt, vttemp);
}
if( nru>0 )
{
applyrotationsfromtheright(!fwddir, ustart, uend, ll+ustart-1, m+ustart-1, work0, work1, u, utemp);
}
if( ncc>0 )
{
applyrotationsfromtheleft(!fwddir, ll+cstart-1, m+cstart-1, cstart, cend, work0, work1, c, ctemp);
}
}
}
//
// QR iteration finished, go back and check convergence
//
continue;
}
//
// All singular values converged, so make them positive
//
for(i = 1; i <= n; i++)
{
if( d(i)<0 )
{
d(i) = -d(i);
//
// Change sign of singular vectors, if desired
//
if( ncvt>0 )
{
ap::vmul(&vt(i+vstart-1, vstart), ap::vlen(vstart,vend), -1);
}
}
}
//
// Sort the singular values into decreasing order (insertion sort on
// singular values, but only one transposition per singular vector)
//
for(i = 1; i <= n-1; i++)
{
//
// Scan for smallest D(I)
//
isub = 1;
smin = d(1);
for(j = 2; j <= n+1-i; j++)
{
if( d(j)<=smin )
{
isub = j;
smin = d(j);
}
}
if( isub!=n+1-i )
{
//
// Swap singular values and vectors
//
d(isub) = d(n+1-i);
d(n+1-i) = smin;
if( ncvt>0 )
{
j = n+1-i;
ap::vmove(&vttemp(vstart), &vt(isub+vstart-1, vstart), ap::vlen(vstart,vend));
ap::vmove(&vt(isub+vstart-1, vstart), &vt(j+vstart-1, vstart), ap::vlen(vstart,vend));
ap::vmove(&vt(j+vstart-1, vstart), &vttemp(vstart), ap::vlen(vstart,vend));
}
if( nru>0 )
{
j = n+1-i;
ap::vmove(utemp.getvector(ustart, uend), u.getcolumn(isub+ustart-1, ustart, uend));
ap::vmove(u.getcolumn(isub+ustart-1, ustart, uend), u.getcolumn(j+ustart-1, ustart, uend));
ap::vmove(u.getcolumn(j+ustart-1, ustart, uend), utemp.getvector(ustart, uend));
}
if( ncc>0 )
{
j = n+1-i;
ap::vmove(&ctemp(cstart), &c(isub+cstart-1, cstart), ap::vlen(cstart,cend));
ap::vmove(&c(isub+cstart-1, cstart), &c(j+cstart-1, cstart), ap::vlen(cstart,cend));
ap::vmove(&c(j+cstart-1, cstart), &ctemp(cstart), ap::vlen(cstart,cend));
}
}
}
return result;
}
