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|
/*
* Copyright (c) 2011, Michael Lehn
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1) Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2) Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
* 3) Neither the name of the FLENS development group nor the names of
* its contributors may be used to endorse or promote products derived
* from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/* Based on
SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
SUBROUTINE ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
$ IHIZ, Z, LDZ, INFO )
*
* -- LAPACK auxiliary routine (version 3.2) --
* Univ. of Tennessee, Univ. of California Berkeley,
* Univ. of Colorado Denver and NAG Ltd..
* November 2006
*/
#ifndef FLENS_LAPACK_LA_LAHQR_TCC
#define FLENS_LAPACK_LA_LAHQR_TCC 1
#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>
namespace flens { namespace lapack {
//== generic lapack implementation =============================================
namespace generic {
//
// Real variant
//
template <typename IndexType, typename MH, typename VWR, typename VWI,
typename MZ>
IndexType
lahqr_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VWR> &wr,
DenseVector<VWI> &wi,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z)
{
using std::abs;
using std::max;
using std::min;
typedef typename GeMatrix<MH>::ElementType T;
const Underscore<IndexType> _;
const T Zero(0), One(1), Two(2);
const T Dat1 = T(3)/T(4),
Dat2 = T(-0.4375);
const IndexType itMax = 30;
const IndexType n = H.numRows();
typedef typename GeMatrix<MH>::VectorView VectorView;
T vBuffer[3];
VectorView v = typename VectorView::Engine(3, vBuffer);
//
// Quick return if possible
//
if (n==0) {
return 0;
}
if (iLo==iHi) {
wr(iLo) = H(iLo, iLo);
wi(iLo) = Zero;
return 0;
}
//
// ==== clear out the trash ====
//
for (IndexType j=iLo; j<=iHi-3; ++j) {
H(j+2, j) = Zero;
H(j+3, j) = Zero;
}
if (iLo<=iHi-2) {
H(iHi, iHi-2) = Zero;
}
const IndexType nh = iHi - iLo + 1;
//
// Set machine-dependent constants for the stopping criterion.
//
T safeMin = lamch<T>(SafeMin);
T safeMax = One / safeMin;
labad(safeMin, safeMax);
const T ulp = lamch<T>(Precision);
const T smallNum = safeMin*(T(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.
//
IndexType i1 = -1, i2 = -1;
if (wantT) {
i1 = 1;
i2 = n;
}
//
// The main loop begins here. Variable 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.
//
IndexType i = iHi;
while (true) {
IndexType l = iLo;
if (i<iLo) {
break;
}
//
// 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.
//
IndexType its;
for (its=0; its<=itMax; ++its) {
//
// Look for a single small subdiagonal element.
//
IndexType k;
for (k=i; k>=l+1; --k) {
if (abs(H(k,k-1))<=smallNum) {
break;
}
T test = abs(H(k-1,k-1)) + abs(H(k,k));
if (test==Zero) {
if (k-2>=iLo) {
test += abs(H(k-1,k-2));
}
if (k+1<=iHi) {
test += abs(H(k+1,k));
}
}
// ==== The following is a conservative small subdiagonal
// . deflation criterion due to Ahues & Tisseur (LAWN 122,
// . 1997). It has better mathematical foundation and
// . improves accuracy in some cases. ====
if (abs(H(k,k-1))<=ulp*test) {
const T ab = max(abs(H(k, k-1)), abs(H(k-1, k)));
const T ba = min(abs(H(k, k-1)), abs(H(k-1, k)));
const T aa = max(abs(H(k, k)), abs(H(k-1, k-1)-H(k, k)));
const T bb = min(abs(H(k, k)), abs(H(k-1, k-1)-H(k, k)));
const T s = aa + ab;
if (ba*(ab/s)<=max(smallNum, ulp*(bb*(aa/s)))) {
break;
}
}
}
l = k;
if (l>iLo) {
//
// H(l,l-1) is negligible
//
H(l, l-1) = Zero;
}
//
// Exit from loop if a submatrix of order 1 or 2 has split off.
//
if (l>=i-1) {
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;
}
T H11, H12, H21, H22;
T rt1r, rt2r, rt1i, rt2i;
if (its==10) {
//
// Exceptional shift.
//
const T s = abs(H(l+1,l)) + abs(H(l+2, l+1));
H11 = Dat1*s + H(l,l);
H12 = Dat2*s;
H21 = s;
H22 = H11;
} else if (its==20) {
//
// Exceptional shift.
//
const T s = abs(H(i,i-1)) + abs(H(i-1,i-2));
H11 = Dat1*s + H(i,i);
H12 = Dat2*s;
H21 = s;
H22 = H11;
} else {
//
// Prepare to use Francis' double shift
// (i.e. 2nd degree generalized Rayleigh quotient)
//
H11 = H(i-1, i-1);
H21 = H( i, i-1);
H12 = H(i-1, i);
H22 = H( i, i);
}
const T s = abs(H11) + abs(H12) + abs(H21) + abs(H22);
if (s==Zero) {
rt1r = Zero;
rt1i = Zero;
rt2r = Zero;
rt2i = Zero;
} else {
H11 /= s;
H21 /= s;
H12 /= s;
H22 /= s;
const T tr = (H11+H22) / Two;
const T det = (H11-tr)*(H22-tr) - H12*H21;
const T rtDisc = sqrt(abs(det));
if (det>=Zero) {
//
// ==== complex conjugate shifts ====
//
rt1r = tr*s;
rt2r = rt1r;
rt1i = rtDisc*s;
rt2i = -rt1i;
} else {
//
// ==== real shifts (use only one of them) ====
//
rt1r = tr + rtDisc;
rt2r = tr - rtDisc;
if (abs(rt1r-H22)<=abs(rt2r-H22)) {
rt1r *= s;
rt2r = rt1r;
} else {
rt2r *= s;
rt1r = rt2r;
}
rt1i = Zero;
rt2i = Zero;
}
}
//
// Look for two consecutive small subdiagonal elements.
//
IndexType m;
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. (The following uses scaling to avoid
// overflows and most underflows.)
//
T H21S = H(m+1,m);
T s = abs(H(m,m)-rt2r) + abs(rt2i) + abs(H21S);
H21S = H(m+1,m)/s;
v(1) = H21S*H(m,m+1)
+ (H(m,m)-rt1r)*((H(m,m)-rt2r)/s)
- rt1i*(rt2i/s);
v(2) = H21S*(H(m,m) + H(m+1,m+1)-rt1r-rt2r);
v(3) = H21S*H(m+2,m+1);
s = abs(v(1)) + abs(v(2)) + abs(v(3));
v(1) /= s;
v(2) /= s;
v(3) /= s;
if (m==l) {
break;
}
const T value1 = abs(H(m,m-1))*(abs(v(2))+abs(v(3)));
const T value2 = ulp*abs(v(1))
*(abs(H(m-1,m-1))+abs(H(m,m))+abs(H(m+1,m+1)));
if (value1<=value2) {
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.
//
T t1, t2, t3, v2, v3;
IndexType nr = min(IndexType(3), i-k+1);
if (k>m) {
v(_(1,nr)) = H(_(k,k+nr-1),k-1);
}
larfg(nr, v(1), v(_(2,nr)), t1);
if (k>m) {
H(k, k-1) = v(1);
H(k+1,k-1) = Zero;
if (k<i-1) {
H(k+2,k-1) = Zero;
}
} else if (m>l) {
// ==== Use the following instead of
// . H( K, K-1 ) = -H( K, K-1 ) to
// . avoid a bug when v(2) and v(3)
// . underflow. ====
H(k, k-1) *= (One-t1);
}
v2 = v(2);
t2 = t1*v2;
if (nr==3) {
v3 = v(3);
t3 = t1*v3;
//
// Apply G from the left to transform the rows of the matrix
// in columns k to i2.
//
for (IndexType j=k; j<=i2; ++j) {
const T sum = H(k,j) + v2*H(k+1,j) + v3*H(k+2,j);
H(k, j) -= sum*t1;
H(k+1, j) -= sum*t2;
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 (IndexType j=i1; j<=min(k+3,i); ++j) {
const T sum = H(j, k) + v2*H(j,k+1) + v3*H(j,k+2);
H(j, k ) -= sum*t1;
H(j, k+1) -= sum*t2;
H(j, k+2) -= sum*t3;
}
if (wantZ) {
//
// Accumulate transformations in the matrix Z
//
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
const T sum = Z(j, k) + v2*Z(j, k+1) + v3*Z(j, k+2);
Z(j, k ) -= sum*t1;
Z(j, k+1) -= sum*t2;
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 (IndexType j=k; j<=i2; ++j) {
const T sum = H(k, j) + v2*H(k+1, j);
H(k, j) -= sum*t1;
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 (IndexType j=i1; j<=i; ++j) {
const T sum = H(j, k) + v2*H(j, k+1);
H(j, k ) -= sum*t1;
H(j, k+1) -= sum*t2;
}
if (wantZ) {
//
// Accumulate transformations in the matrix Z
//
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
const T sum = Z(j, k) + v2*Z(j, k+1);
Z(j, k ) -= sum*t1;
Z(j, k+1) -= sum*t2;
}
}
}
}
}
//
// Failure to converge in remaining number of iterations
//
if (its>itMax) {
return i;
}
if (l==i) {
//
// H(I,I-1) is negligible: one eigenvalue has converged.
//
wr(i) = H(i, i);
wi(i) = Zero;
} 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.
//
T cs, sn;
lanv2(H(i-1,i-1), H(i-1,i), H(i,i-1), H(i,i),
wr(i-1), wi(i-1), wr(i), wi(i),
cs, sn);
if (wantT) {
//
// Apply the transformation to the rest of H.
//
if (i2>i) {
const auto cols = _(i+1,i2);
blas::rot(H(i-1,cols), H(i,cols), cs, sn);
}
const auto rows = _(i1,i-2);
blas::rot(H(rows, i-1), H(rows, i), cs, sn);
}
if (wantZ) {
//
// Apply the transformation to Z.
//
const auto rows = _(iLoZ, iHiZ);
blas::rot(Z(rows, i-1), Z(rows, i), cs, sn);
}
}
//
// return to start of the main loop with new value of I.
//
i = l - 1;
}
return 0;
}
//
// Complex variant
//
template <typename IndexType, typename MH, typename VW, typename MZ>
IndexType
lahqr_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VW> &w,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z)
{
using std::abs;
using std::conj;
using std::imag;
using std::max;
using std::min;
using std::real;
typedef typename GeMatrix<MH>::ElementType T;
typedef typename ComplexTrait<T>::PrimitiveType PT;
const Underscore<IndexType> _;
const T Zero(0), One(1);
const PT RZero(0), ROne(1), RHalf(0.5);
const PT Dat1 = PT(3)/PT(4);
const IndexType itMax = 30;
const IndexType n = H.numRows();
typedef typename GeMatrix<MH>::VectorView VectorView;
T vBuffer[2];
VectorView v = typename VectorView::Engine(2, vBuffer);
//
// Quick return if possible
//
if (n==0) {
return 0;
}
if (iLo==iHi) {
w(iLo) = H(iLo, iLo);
return 0;
}
//
// ==== clear out the trash ====
//
for (IndexType j=iLo; j<=iHi-3; ++j) {
H(j+2, j) = Zero;
H(j+3, j) = Zero;
}
if (iLo<=iHi-2) {
H(iHi, iHi-2) = Zero;
}
//
// ==== ensure that subdiagonal entries are real ====
//
const IndexType jLo = (wantT) ? 1 : iLo;
const IndexType jHi = (wantT) ? n : iHi;
for (IndexType i=iLo+1; i<=iHi; ++i) {
if (imag(H(i,i-1))!=RZero) {
// ==== The following redundant normalization
// . avoids problems with both gradual and
// . sudden underflow in ABS(H(I,I-1)) ====
T scale = H(i,i-1) / abs1(H(i,i-1));
scale = conj(scale) / abs(scale);
H(i,i-1) = abs(H(i,i-1));
H(i, _(i,jHi)) *= scale;
H(_(jLo,min(jHi,i+1)), i) *= conj(scale);
if (wantZ) {
Z(_(iLoZ,iHiZ),i) *= conj(scale);
}
}
}
const IndexType nh = iHi - iLo + 1;
//
// Set machine-dependent constants for the stopping criterion.
//
PT safeMin = lamch<PT>(SafeMin);
PT safeMax = ROne / safeMin;
labad(safeMin, safeMax);
const PT ulp = lamch<PT>(Precision);
const PT smallNum = safeMin*(PT(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.
//
IndexType i1 = -1, i2 = -1;
if (wantT) {
i1 = 1;
i2 = n;
}
//
// The main loop begins here. Variable 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.
//
IndexType i = iHi;
while (true) {
if (i<iLo) {
break;
}
//
// 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.
//
IndexType l = iLo;
IndexType its;
for (its=0; its<=itMax; ++its) {
//
// Look for a single small subdiagonal element.
//
IndexType k;
for (k=i; k>=l+1; --k) {
if (abs1(H(k,k-1))<=smallNum) {
break;
}
PT test = abs1(H(k-1,k-1)) + abs1(H(k,k));
if (test==RZero) {
if (k-2>=iLo) {
test += abs(real(H(k-1,k-2)));
}
if (k+1<=iHi) {
test += abs(real(H(k+1,k)));
}
}
// ==== The following is a conservative small subdiagonal
// . deflation criterion due to Ahues & Tisseur (LAWN 122,
// . 1997). It has better mathematical foundation and
// . improves accuracy in some cases. ====
if (abs(real(H(k,k-1)))<=ulp*test) {
const PT ab = max(abs1(H(k, k-1)),
abs1(H(k-1, k)));
const PT ba = min(abs1(H(k, k-1)),
abs1(H(k-1, k)));
const PT aa = max(abs1(H(k, k)),
abs1(H(k-1, k-1)-H(k, k)));
const PT bb = min(abs1(H(k, k)),
abs1(H(k-1, k-1)-H(k, k)));
const PT s = aa + ab;
if (ba*(ab/s)<=max(smallNum, ulp*(bb*(aa/s)))) {
break;
}
}
}
l = k;
if (l>iLo) {
//
// H(l,l-1) is negligible
//
H(l, l-1) = Zero;
}
//
// Exit from loop if a submatrix of order 1 or 2 has split off.
//
if (l>=i) {
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;
}
T H11, H11S, H22, t, u, x, y;
PT H10, H21, s, sx;
if (its==10) {
//
// Exceptional shift.
//
s = Dat1*abs(real(H(l+1,l)));
t = s + H(l,l);
} else if (its==20) {
//
// Exceptional shift.
//
s = Dat1*abs(real(H(i,i-1)));
t = s + H(i,i);
} else {
//
//
// Wilkinson's shift.
//
//
t = H(i,i);
u = sqrt(H(i-1,i)) * sqrt(H(i,i-1));
s = abs1(u);
if (s != RZero) {
x = RHalf * (H(i-1,i-1) - t);
sx = abs1(x);
s = max(s, abs1(x));
y = s * sqrt(pow(x/s,2) + pow(u/s,2));
if (sx > RZero) {
PT tmp = real(x/sx)*real(y)
+ imag(x/sx)*imag(y);
if (tmp<RZero) {
y = -y;
}
}
t -= u*ladiv(u, x+y);
}
}
//
// Look for two consecutive small subdiagonal elements.
//
IndexType m;
for (m=i-1; m>=l+1; --m) {
//
// Determine the effect of starting the single-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);
H11S = H11 - t;
H21 = real(H(m+1,m));
s = abs1(H11S) + abs(H21);
H11S /= s;
H21 /= s;
v(1) = H11S;
v(2) = H21;
H10 = real(H(m,m-1));
if (abs(H10)*abs(H21)<=ulp*(abs1(H11S)*(abs1(H11)+abs1(H22)))) {
goto SINGLE_SHIFT;
}
}
H11 = H(l, l);
H22 = H(l+1, l+1);
H11S = H11 - t;
H21 = real(H(l+1,l));
s = abs1(H11S) + abs(H21);
H11S /= s;
H21 /= s;
v(1) = H11S;
v(2) = H21;
SINGLE_SHIFT:
//
// Single-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.
//
// V(2) is always real before the call to ZLARFG, and hence
// after the call T2 ( = T1*V(2) ) is also real.
//
T t1, v2;
PT t2;
if (k>m) {
v = H(_(k,k+1),k-1);
}
larfg(2, v(1), v(_(2,2)), t1);
if (k>m) {
H(k, k-1) = v(1);
H(k+1,k-1) = Zero;
}
v2 = v(2);
t2 = real(t1*v2);
//
// Apply G from the left to transform the rows of the matrix
// in columns K to I2.
//
for (IndexType j=k; j<=i2; ++j) {
const T sum = conj(t1)*H(k,j) + t2*H(k+1,j);
H(k, j) -= sum;
H(k+1,j) -= sum*v2;
}
//
// Apply G from the right to transform the columns of the
// matrix in rows I1 to min(K+2,I).
//
for (IndexType j=i1; j<=min(k+2,i); ++j) {
T sum = t1*H(j,k) + t2*H(j,k+1);
H(j,k) -= sum;
H(j,k+1) -= sum*conj(v2);
}
if (wantZ) {
//
// Accumulate transformations in the matrix Z
//
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
T sum = t1*Z(j,k) + t2*Z(j,k+1);
Z(j,k) -= sum;
Z(j,k+1) -= sum*conj(v2);
}
}
if (k==m && m>l) {
//
// If the QR step was started at row M > L because two
// consecutive small subdiagonals were found, then extra
// scaling must be performed to ensure that H(M,M-1) remains
// real.
//
T tmp = One - t1;
tmp /= abs(tmp);
H(m+1,m) *= conj(tmp);
if (m+2 <= i) {
H(m+2,m+1) *= tmp;
}
for (IndexType j=m; j<=i; ++j) {
if (j != m+1) {
if (i2>j) {
H(j,_(j+1,i2)) *= tmp;
}
H(_(i1,j-1),j) *= conj(tmp);
if (wantZ) {
Z(_(iLoZ,iHiZ),j) *= conj(tmp);
}
}
}
}
}
//
// Ensure that H(I,I-1) is real.
//
T tmp = H(i,i-1);
if (imag(tmp) != RZero) {
PT rTmp = abs(tmp);
H(i,i-1) = rTmp;
tmp /= rTmp;
if (i2>i) {
H(i,_(i+1,i2)) *= conj(tmp);
}
H(_(i1,i-1),i) *= tmp;
if (wantZ) {
Z(_(iLoZ,iHiZ),i) *= tmp;
}
}
}
//
// Failure to converge in remaining number of iterations
//
if (its>itMax) {
return i;
}
//
// H(I,I-1) is negligible: one eigenvalue has converged.
//
w(i) = H(i,i);
//
// return to start of the main loop with new value of I.
//
i = l - 1;
}
return 0;
}
} // namespace generic
//== interface for native lapack ===============================================
#ifdef USE_CXXLAPACK
namespace external {
//
// Real variant
//
template <typename IndexType, typename MH, typename VWR, typename VWI,
typename MZ>
IndexType
lahqr_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VWR> &wr,
DenseVector<VWI> &wi,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z)
{
IndexType info;
info = cxxlapack::lahqr<IndexType>(wantT,
wantZ,
H.numRows(),
iLo,
iHi,
H.data(),
H.leadingDimension(),
wr.data(),
wi.data(),
iLoZ,
iHiZ,
Z.data(),
Z.leadingDimension());
return info;
}
//
// Complex variant
//
template <typename IndexType, typename MH, typename VW, typename MZ>
IndexType
lahqr_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VW> &w,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z)
{
IndexType info;
info = cxxlapack::lahqr<IndexType>(wantT,
wantZ,
H.numRows(),
iLo,
iHi,
H.data(),
H.leadingDimension(),
w.data(),
iLoZ,
iHiZ,
Z.data(),
Z.leadingDimension());
return info;
}
} // namespace external
#endif // USE_CXXLAPACK
//== public interface ==========================================================
//
// Real variant
//
template <typename IndexType, typename MH, typename VWR, typename VWI,
typename MZ>
typename RestrictTo<IsRealGeMatrix<MH>::value
&& IsRealDenseVector<VWR>::value
&& IsRealDenseVector<VWI>::value
&& IsRealGeMatrix<MZ>::value,
IndexType>::Type
lahqr(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
MH &&H,
VWR &&wr,
VWI &&wi,
IndexType iLoZ,
IndexType iHiZ,
MZ &&Z)
{
LAPACK_DEBUG_OUT("lahqr");
//
// Remove references from rvalue types
//
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<MH>::Type MatrixH;
typedef typename RemoveRef<VWR>::Type VectorWR;
typedef typename RemoveRef<VWI>::Type VectorWI;
typedef typename RemoveRef<MZ>::Type MatrixZ;
# endif
//
// Test the input parameters
//
using std::max;
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
ASSERT(wr.firstIndex()==1);
ASSERT(wr.length()==H.numRows());
ASSERT(wi.firstIndex()==1);
ASSERT(wi.length()==H.numRows());
ASSERT(wantZ || (Z.numRows()==H.numCols()));
ASSERT(wantZ || (Z.numCols()==H.numCols()));
// 1 <= ILO <= max(1,IHI); IHI <= N.
ASSERT(1<=iLo);
ASSERT(iLo<=max(IndexType(1), iHi));
ASSERT(iHi<=H.numRows());
// 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
ASSERT(1<=iLoZ);
ASSERT(iLoZ<=iLo);
ASSERT(iHi<=iHiZ);
ASSERT(iHiZ<=H.numRows());
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename MatrixH::NoView H_org = H;
typename MatrixZ::NoView Z_org = Z;
typename MatrixH::NoView H_ = H;
typename VectorWR::NoView wr_ = wr;
typename VectorWI::NoView wi_ = wi;
typename MatrixZ::NoView Z_ = Z;
# endif
//
// Call implementation
//
IndexType info = LAPACK_SELECT::lahqr_impl(wantT, wantZ, iLo, iHi,
H, wr, wi, iLoZ, iHiZ, Z);
//
// Compare results
//
# ifdef CHECK_CXXLAPACK
IndexType info_ = external::lahqr_impl(wantT, wantZ, iLo, iHi,
H_, wr_, wi_, iLoZ, iHiZ, Z_);
bool failed = false;
if (! isIdentical(H, H_, " H", "H_")) {
std::cerr << "CXXLAPACK: H = " << H << std::endl;
std::cerr << "F77LAPACK: H_ = " << H_ << std::endl;
failed = true;
}
if (! isIdentical(wr, wr_, " wr", "wr_")) {
std::cerr << "CXXLAPACK: wr = " << wr << std::endl;
std::cerr << "F77LAPACK: wr_ = " << wr_ << std::endl;
failed = true;
}
if (! isIdentical(wi, wi_, " wi", "wi_")) {
std::cerr << "CXXLAPACK: wi = " << wi << std::endl;
std::cerr << "F77LAPACK: wi_ = " << wi_ << std::endl;
failed = true;
}
if (! isIdentical(Z, Z_, " Z", "Z_")) {
std::cerr << "CXXLAPACK: Z = " << Z << std::endl;
std::cerr << "F77LAPACK: Z_ = " << Z_ << std::endl;
failed = true;
}
if (! isIdentical(info, info_, " info", "info_")) {
std::cerr << "CXXLAPACK: info = " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << std::endl;
failed = true;
}
if (failed) {
std::cerr << "H_org = " << H_org << std::endl;
std::cerr << "Z_org = " << Z_org << std::endl;
std::cerr << "wantT = " << wantT
<< ", wantZ = " << wantZ
<< ", iLo = " << iLo
<< ", iHi = " << iHi
<< std::endl;
std::cerr << "error in: lahqr.tcc" << std::endl;
ASSERT(0);
} else {
// std::cerr << "passed: lahqr.tcc" << std::endl;
}
# endif
return info;
}
//
// Complex variant
//
template <typename IndexType, typename MH, typename VW, typename MZ>
typename RestrictTo<IsComplexGeMatrix<MH>::value
&& IsComplexDenseVector<VW>::value
&& IsComplexGeMatrix<MZ>::value,
IndexType>::Type
lahqr(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
MH &&H,
VW &&w,
IndexType iLoZ,
IndexType iHiZ,
MZ &&Z)
{
LAPACK_DEBUG_OUT("lahqr (complex)");
//
// Remove references from rvalue types
//
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<MH>::Type MatrixH;
typedef typename RemoveRef<VW>::Type VectorW;
typedef typename RemoveRef<MZ>::Type MatrixZ;
# endif
//
// Test the input parameters
//
using std::max;
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
ASSERT(w.firstIndex()==1);
ASSERT(w.length()==H.numRows());
ASSERT(wantZ || (Z.numRows()==H.numCols()));
ASSERT(wantZ || (Z.numCols()==H.numCols()));
// 1 <= ILO <= max(1,IHI); IHI <= N.
ASSERT(1<=iLo);
ASSERT(iLo<=max(IndexType(1), iHi));
ASSERT(iHi<=H.numRows());
// 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
ASSERT(1<=iLoZ);
ASSERT(iLoZ<=iLo);
ASSERT(iHi<=iHiZ);
ASSERT(iHiZ<=H.numRows());
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename MatrixH::NoView H_org = H;
typename MatrixZ::NoView Z_org = Z;
typename MatrixH::NoView H_ = H;
typename VectorW::NoView w_ = w;
typename MatrixZ::NoView Z_ = Z;
# endif
//
// Call implementation
//
IndexType info = LAPACK_SELECT::lahqr_impl(wantT, wantZ, iLo, iHi,
H, w, iLoZ, iHiZ, Z);
//
// Compare results
//
# ifdef CHECK_CXXLAPACK
IndexType info_ = external::lahqr_impl(wantT, wantZ, iLo, iHi,
H_, w_, iLoZ, iHiZ, Z_);
bool failed = false;
if (! isIdentical(H, H_, " H", "H_")) {
std::cerr << "CXXLAPACK: H = " << H << std::endl;
std::cerr << "F77LAPACK: H_ = " << H_ << std::endl;
failed = true;
}
if (! isIdentical(w, w_, " w", "w_")) {
std::cerr << "CXXLAPACK: w = " << w << std::endl;
std::cerr << "F77LAPACK: w_ = " << w_ << std::endl;
failed = true;
}
if (! isIdentical(Z, Z_, " Z", "Z_")) {
std::cerr << "CXXLAPACK: Z = " << Z << std::endl;
std::cerr << "F77LAPACK: Z_ = " << Z_ << std::endl;
failed = true;
}
if (! isIdentical(info, info_, " info", "info_")) {
std::cerr << "CXXLAPACK: info = " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << std::endl;
failed = true;
}
if (failed) {
std::cerr << "H_org = " << H_org << std::endl;
std::cerr << "Z_org = " << Z_org << std::endl;
std::cerr << "wantT = " << wantT
<< ", wantZ = " << wantZ
<< ", iLo = " << iLo
<< ", iHi = " << iHi
<< std::endl;
std::cerr << "error in: lahqr.tcc" << std::endl;
ASSERT(0);
} else {
// std::cerr << "passed: lahqr.tcc" << std::endl;
}
# endif
return info;
}
} } // namespace lapack, flens
#endif // FLENS_LAPACK_LA_LAHQR_TCC
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