/*
Diagonalization of an mxm matrix A by a rotation R.
*/
int Diago(double *A, double *R, int m, double threshold)
{
int encore = 1;
int rots = 0;
int p, q;
double theta, c, s;
Identity(R, m);
/* Sweeps until no pair gets updated */
while (encore>0) {
encore = 0;
for (p = 0; p<m; p++)
for (q = p + 1; q<m; q++) {
theta = Givens(A, m, p, q);
if (fabs(theta) > threshold) {
c = cos(theta);
s = sin(theta);
LeftRotSimple(A, m, m, p, q, c, s);
RightRotSimple(A, m, m, p, q, c, s);
LeftRotSimple(R, m, m, p, q, c, s);
encore = 1;
rots++;
}
}
}
return rots;
}
void NeighborColSwap
( Matrix<F>& Q,
Matrix<F>& R,
Int j )
{
DEBUG_CSE
typedef Base<F> Real;
const Int m = Q.Height();
ColSwap( R, j, j+1 );
if( j < m-1 )
{
Real c; F s;
Givens( R(j,j), R(j+1,j), c, s );
auto RBR = R( IR(j,END), IR(j,END) );
// RBR((0,1),:) := | c, s | RBR((0,1),:)
// | -conj(s), c |
RotateRows( c, s, RBR, 0, 1 );
// Since | c, s |^H = | c, -s |, the transformation
// | -conj(s), c | | conj(s), c |
//
// s |-> -s inverts a Givens rotation matrix.
//
// We therefore update
//
// Q(:,(j,j+1)) := Q(:,(j,j+1)) | c, -s |.
// | conj(s), c |
RotateCols( c, -s, Q, j, j+1 );
}
}
//#####################################################################
// Function Implicit_QR_Step
//#####################################################################
template<class T,int bandwidth> void BANDED_SYMMETRIC_MATRIX<T,bandwidth>::
Implicit_QR_Step(const int start,const T shift)
{
// see Golub and Van Loan, 8.3.2, p. 420
T x=A(start)[1]-shift,z=A(start)[2];
for(int k=start;;k++){ // chase offdiagonal element z into oblivion
T c,s;Givens(x,z,c,s);
if(k>start){T& b=A(k-1)[2];b=c*b-s*z;}
T &ap=A(k)[1],&bp=A(k)[2],&aq=A(k+1)[1],&bq=A(k+1)[2]; // p,q = k,k+1
T cc=c*c,cs=c*s,ss=s*s,ccap=cc*ap,csbp=cs*bp,ssaq=ss*aq,csap=cs*ap,ccbp=cc*bp,ssbp=ss*bp,csaq=cs*aq,ssap=ss*ap,ccaq=cc*aq;
ap=ccap-2*csbp+ssaq;bp=csap+ccbp-ssbp-csaq;aq=ssap+2*csbp+ccaq; // TODO: save 7 flops somehow
if(!bq) return; // stop if matrix decouples
x=bp;z=-s*bq;bq=c*bq;}
}
void NeighborColSwap
( Matrix<F>& Q,
Matrix<F>& R,
Int j )
{
DEBUG_CSE
typedef Base<F> Real;
const Int m = Q.Height();
ColSwap( R, j, j+1 );
if( j < m-1 )
{
Real c; F s;
Givens( R.Get(j,j), R.Get(j+1,j), c, s );
auto RBR = R( IR(j,END), IR(j,END) );
RotateRows( c, s, RBR, 0, 1 );
auto col1 = Q( ALL, IR(j) );
auto col2 = Q( ALL, IR(j+1) );
RotateCols( c, Conj(s), Q, j, j+1 );
}
}
void LUTUnb
( bool conjugate,
const Matrix<F>& U,
Matrix<F>& X,
bool checkIfSingular )
{
EL_DEBUG_CSE
typedef Base<F> Real;
const Int m = X.Height();
const Int n = X.Width();
if( conjugate )
Conjugate( X );
const F* UBuf = U.LockedBuffer();
F* XBuf = X.Buffer();
const Int ldu = U.LDim();
const Int ldx = X.LDim();
Int k=0;
while( k < m )
{
const bool in2x2 = ( k+1<m && UBuf[(k+1)+k*ldu] != F(0) );
if( in2x2 )
{
// Solve the 2x2 linear systems via a 2x2 QR decomposition produced
// by the Givens rotation
// | c s | | U(k, k) | = | gamma11 |
// | -conj(s) c | | U(k+1,k) | | 0 |
//
// and by also forming the right two entries of the 2x2 resulting
// upper-triangular matrix, say gamma12 and gamma22
//
// Extract the 2x2 diagonal block, D
const F delta11 = UBuf[ k + k *ldu];
const F delta12 = UBuf[ k +(k+1)*ldu];
const F delta21 = UBuf[(k+1)+ k *ldu];
const F delta22 = UBuf[(k+1)+(k+1)*ldu];
// Decompose D = Q R
Real c; F s;
const F gamma11 = Givens( delta11, delta21, c, s );
const F gamma12 = c*delta12 + s*delta22;
const F gamma22 = -Conj(s)*delta12 + c*delta22;
if( checkIfSingular )
{
// TODO: Check if sufficiently small instead
if( gamma11 == F(0) || gamma22 == F(0) )
LogicError("Singular diagonal block detected");
}
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve against R^T
xBuf[k ] /= gamma11;
xBuf[k+1] -= gamma12*xBuf[k];
xBuf[k+1] /= gamma22;
// Solve against Q^T
const F chi1 = xBuf[k ];
const F chi2 = xBuf[k+1];
xBuf[k ] = c*chi1 - Conj(s)*chi2;
xBuf[k+1] = s*chi1 + c*chi2;
// Update x2 := x2 - U12^T x1
blas::Axpy
( m-(k+2), -xBuf[k ],
&UBuf[ k +(k+2)*ldu], ldu, &xBuf[k+2], 1 );
blas::Axpy
( m-(k+2), -xBuf[k+1],
&UBuf[(k+1)+(k+2)*ldu], ldu, &xBuf[k+2], 1 );
}
k += 2;
}
else
{
if( checkIfSingular )
{
// TODO: Check if sufficiently small instead
if( UBuf[k+k*ldu] == F(0) )
LogicError("Singular diagonal entry detected");
}
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
xBuf[k] /= UBuf[k+k*ldu];
blas::Axpy
( m-(k+1), -xBuf[k], &UBuf[k+(k+1)*ldu], ldu, &xBuf[k+1], 1 );
}
k += 1;
}
}
if( conjugate )
Conjugate( X );
}
void LLTUnb
( bool conjugate, const Matrix<F>& L, Matrix<F>& X, bool checkIfSingular )
{
EL_DEBUG_CSE
typedef Base<F> Real;
const Int m = X.Height();
const Int n = X.Width();
const F* LBuf = L.LockedBuffer();
F* XBuf = X.Buffer();
const Int ldl = L.LDim();
const Int ldx = X.LDim();
if( conjugate )
Conjugate( X );
Int k=m-1;
while( k >= 0 )
{
const bool in2x2 = ( k>0 && LBuf[(k-1)+k*ldl] != F(0) );
if( in2x2 )
{
--k;
// Solve the 2x2 linear systems via a 2x2 LQ decomposition produced
// by the Givens rotation
// | L(k,k) L(k,k+1) | | c -conj(s) | = | gamma11 0 |
// | s c |
// and by also forming the bottom two entries of the 2x2 resulting
// lower-triangular matrix, say gamma21 and gamma22
//
// Extract the 2x2 diagonal block, D
const F delta11 = LBuf[ k + k *ldl];
const F delta12 = LBuf[ k +(k+1)*ldl];
const F delta21 = LBuf[(k+1)+ k *ldl];
const F delta22 = LBuf[(k+1)+(k+1)*ldl];
// Decompose D = L Q
Real c; F s;
const F gamma11 = Givens( delta11, delta12, c, s );
const F gamma21 = c*delta21 + s*delta22;
const F gamma22 = -Conj(s)*delta21 + c*delta22;
if( checkIfSingular )
{
// TODO: Instead check if values are too small in magnitude
if( gamma11 == F(0) || gamma22 == F(0) )
LogicError("Singular diagonal block detected");
}
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve against Q^T
const F chi1 = xBuf[k ];
const F chi2 = xBuf[k+1];
xBuf[k ] = c*chi1 + s*chi2;
xBuf[k+1] = -Conj(s)*chi1 + c*chi2;
// Solve against R^T
xBuf[k+1] /= gamma22;
xBuf[k ] -= gamma21*xBuf[k+1];
xBuf[k ] /= gamma11;
// Update x0 := x0 - L10^T x1
blas::Axpy( k, -xBuf[k ], &LBuf[k ], ldl, xBuf, 1 );
blas::Axpy( k, -xBuf[k+1], &LBuf[k+1], ldl, xBuf, 1 );
}
}
else
{
if( checkIfSingular )
if( LBuf[k+k*ldl] == F(0) )
LogicError("Singular diagonal entry detected");
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve the 1x1 linear system
xBuf[k] /= LBuf[k+k*ldl];
// Update x0 := x0 - l10^T chi_1
blas::Axpy( k, -xBuf[k], &LBuf[k], ldl, xBuf, 1 );
}
}
--k;
}
if( conjugate )
Conjugate( X );
}
Matrix DivideAndSolve(Matrix A,int p)
{
double a,b,d,e,s,c,mu,i;
int h,j,k,l,m,o,rowstartt,colstartt,rowendt,colendt;
Vector v,u,x,y,z;
Matrix B,C,D,E,Q,U,H;
U = newIdMatrix();
H = newMatrix();
v = newVector();
i = p+1;
rowstartt = i;
colstartt = 0;
while (rowstartt<n)
{
rowendt = (rowstartt+i-1<n-1)?rowstartt+i-1:n-1;
colstartt = rowstartt-i;
colendt = colstartt+i-1;
/* Now T_i has dimension (p-h)x(p-h) */
/* First zero all but row 1 in T_i */
for (m=colstartt;m<=colendt;m++)
{
if (norm(A,m,rowstartt,rowendt)!=0.0)
{
/* Find Householder(A(h:p,m)) */
House(A,v,m,rowstartt,rowendt);
for (o=0;o<rowstartt;o++)
v[o]=0.0;
for (o=rowendt+1;o<n;o++)
v[o]=0.0;
ApplyHouse(A,v,rowstartt,rowendt);
}
printf("m=%i, rowstart=%i, rowend=%i\n",m,rowstartt,rowendt);
printVector(v);
printMatrix(A);
}
/* Now zero all but the last entry in row 1 */
WeirdHouse(A,v,rowstartt,colstartt,colendt);
/* Apply the HouseHolder */
ApplyHouse(A,v,colstartt,colendt);
/* Now T_i has one single non-zero entry */
/* Iterate with explicit shift to zero the last
non-zero entry */
while (A[rowstartt][colendt]>
(A[rowstartt-1][colendt]-A[rowstartt][colendt+1])*epsilon)
{
printMatrix(A);
/* Wilkonson Shift?? */
d =(A[rowstartt-1][colendt]-A[rowstartt][colendt+1])/2.0;
b = A[rowstartt][colendt];
mu = A[rowstartt][colendt+1]+d-sign(d)*sqrt(d*d+b*b);
/* Determine the givens */
Givens(A[rowstartt-1][colendt]-mu,
A[rowstartt][colendt],&s,&c);
/* Apply the givens */
ApplyGivens(A,s,c,rowstartt-1,rowstartt,0,n-1);
printf("%e\n",A[colstartt][colendt]);
}
rowstartt+=i;
colstartt+=i;
}
}
void LUTUnb
( bool conjugate,
const Matrix<Real>& U,
const Matrix<Complex<Real>>& shifts,
Matrix<Real>& XReal,
Matrix<Real>& XImag )
{
DEBUG_CSE
typedef Complex<Real> C;
const Int m = XReal.Height();
const Int n = XReal.Width();
if( conjugate )
XImag *= -1;
const Real* UBuf = U.LockedBuffer();
Real* XRealBuf = XReal.Buffer();
Real* XImagBuf = XImag.Buffer();
const Int ldU = U.LDim();
const Int ldXReal = XReal.LDim();
const Int ldXImag = XImag.LDim();
Int k=0;
while( k < m )
{
const bool in2x2 = ( k+1<m && UBuf[(k+1)+k*ldU] != Real(0) );
if( in2x2 )
{
// Solve the 2x2 linear systems via 2x2 QR decompositions produced
// by the Givens rotation
// | c s | | U(k, k)-shift | = | gamma11 |
// | -conj(s) c | | U(k+1,k) | | 0 |
//
// and by also forming the right two entries of the 2x2 resulting
// upper-triangular matrix, say gamma12 and gamma22
//
// Extract the constant part of the 2x2 diagonal block, D
const Real delta12 = UBuf[ k +(k+1)*ldU];
const Real delta21 = UBuf[(k+1)+ k *ldU];
for( Int j=0; j<n; ++j )
{
const C delta11 = UBuf[ k + k *ldU] - shifts.Get(j,0);
const C delta22 = UBuf[(k+1)+(k+1)*ldU] - shifts.Get(j,0);
// Decompose D = Q R
Real c; C s;
const C gamma11 = Givens( delta11, C(delta21), c, s );
const C gamma12 = c*delta12 + s*delta22;
const C gamma22 = -Conj(s)*delta12 + c*delta22;
Real* xRealBuf = &XRealBuf[j*ldXReal];
Real* xImagBuf = &XImagBuf[j*ldXImag];
// Solve against R^T
C chi1(xRealBuf[k ],xImagBuf[k ]);
C chi2(xRealBuf[k+1],xImagBuf[k+1]);
chi1 /= gamma11;
chi2 -= gamma12*chi1;
chi2 /= gamma22;
// Solve against Q^T
const C eta1 = c*chi1 - Conj(s)*chi2;
const C eta2 = s*chi1 + c*chi2;
xRealBuf[k ] = eta1.real();
xImagBuf[k ] = eta1.imag();
xRealBuf[k+1] = eta2.real();
xImagBuf[k+1] = eta2.imag();
// Update x2 := x2 - U12^T x1
blas::Axpy
( m-(k+2), -xRealBuf[k ],
&UBuf[ k +(k+2)*ldU], ldU, &xRealBuf[k+2], 1 );
blas::Axpy
( m-(k+2), -xImagBuf[k ],
&UBuf[ k +(k+2)*ldU], ldU, &xImagBuf[k+2], 1 );
blas::Axpy
( m-(k+2), -xRealBuf[k+1],
&UBuf[(k+1)+(k+2)*ldU], ldU, &xRealBuf[k+2], 1 );
blas::Axpy
( m-(k+2), -xImagBuf[k+1],
&UBuf[(k+1)+(k+2)*ldU], ldU, &xImagBuf[k+2], 1 );
}
k += 2;
}
else
{
for( Int j=0; j<n; ++j )
{
Real* xRealBuf = &XRealBuf[j*ldXReal];
Real* xImagBuf = &XImagBuf[j*ldXImag];
C eta1( xRealBuf[k], xImagBuf[k] );
eta1 /= UBuf[k+k*ldU] - shifts.Get(j,0);
xRealBuf[k] = eta1.real();
xImagBuf[k] = eta1.imag();
blas::Axpy
( m-(k+1), -xRealBuf[k],
&UBuf[k+(k+1)*ldU], ldU, &xRealBuf[k+1], 1 );
blas::Axpy
( m-(k+1), -xImagBuf[k],
&UBuf[k+(k+1)*ldU], ldU, &xImagBuf[k+1], 1 );
}
k += 1;
}
}
if( conjugate )
XImag *= -1;
}
void LUTUnb
( bool conjugate, const Matrix<F>& U, const Matrix<F>& shifts, Matrix<F>& X )
{
DEBUG_CSE
typedef Base<F> Real;
const Int m = X.Height();
const Int n = X.Width();
if( conjugate )
Conjugate( X );
const F* UBuf = U.LockedBuffer();
F* XBuf = X.Buffer();
const Int ldU = U.LDim();
const Int ldX = X.LDim();
Int k=0;
while( k < m )
{
const bool in2x2 = ( k+1<m && UBuf[(k+1)+k*ldU] != F(0) );
if( in2x2 )
{
// Solve the 2x2 linear systems via 2x2 QR decompositions produced
// by the Givens rotation
// | c s | | U(k, k)-shift | = | gamma11 |
// | -conj(s) c | | U(k+1,k) | | 0 |
//
// and by also forming the right two entries of the 2x2 resulting
// upper-triangular matrix, say gamma12 and gamma22
//
// Extract the constant part of the 2x2 diagonal block, D
const F delta12 = UBuf[ k +(k+1)*ldU];
const F delta21 = UBuf[(k+1)+ k *ldU];
for( Int j=0; j<n; ++j )
{
const F delta11 = UBuf[ k + k *ldU] - shifts.Get(j,0);
const F delta22 = UBuf[(k+1)+(k+1)*ldU] - shifts.Get(j,0);
// Decompose D = Q R
Real c; F s;
const F gamma11 = Givens( delta11, delta21, c, s );
const F gamma12 = c*delta12 + s*delta22;
const F gamma22 = -Conj(s)*delta12 + c*delta22;
F* xBuf = &XBuf[j*ldX];
// Solve against R^T
F chi1 = xBuf[k ];
F chi2 = xBuf[k+1];
chi1 /= gamma11;
chi2 -= gamma12*chi1;
chi2 /= gamma22;
// Solve against Q^T
xBuf[k ] = c*chi1 - Conj(s)*chi2;
xBuf[k+1] = s*chi1 + c*chi2;
// Update x2 := x2 - U12^T x1
blas::Axpy
( m-(k+2), -xBuf[k ],
&UBuf[ k +(k+2)*ldU], ldU, &xBuf[k+2], 1 );
blas::Axpy
( m-(k+2), -xBuf[k+1],
&UBuf[(k+1)+(k+2)*ldU], ldU, &xBuf[k+2], 1 );
}
k += 2;
}
else
{
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldX];
xBuf[k] /= UBuf[k+k*ldU] - shifts.Get(j,0);
blas::Axpy
( m-(k+1), -xBuf[k], &UBuf[k+(k+1)*ldU], ldU, &xBuf[k+1], 1 );
}
k += 1;
}
}
if( conjugate )
Conjugate( X );
}
void LUNUnb( const Matrix<F>& U, Matrix<F>& X, bool checkIfSingular )
{
DEBUG_CSE
const Int m = X.Height();
const Int n = X.Width();
typedef Base<F> Real;
const F* UBuf = U.LockedBuffer();
F* XBuf = X.Buffer();
const Int ldu = U.LDim();
const Int ldx = X.LDim();
Int k=m-1;
while( k >= 0 )
{
const bool in2x2 = ( k>0 && UBuf[k+(k-1)*ldu] != F(0) );
if( in2x2 )
{
--k;
// Solve the 2x2 linear systems via a 2x2 QR decomposition produced
// by the Givens rotation
// | c s | | U(k, k) | = | gamma11 |
// | -conj(s) c | | U(k+1,k) | | 0 |
//
// and by also forming the right two entries of the 2x2 resulting
// upper-triangular matrix, say gamma12 and gamma22
//
// Extract the 2x2 diagonal block, D
const F delta11 = UBuf[ k + k *ldu];
const F delta12 = UBuf[ k +(k+1)*ldu];
const F delta21 = UBuf[(k+1)+ k *ldu];
const F delta22 = UBuf[(k+1)+(k+1)*ldu];
// Decompose D = Q R
Real c; F s;
const F gamma11 = Givens( delta11, delta21, c, s );
const F gamma12 = c*delta12 + s*delta22;
const F gamma22 = -Conj(s)*delta12 + c*delta22;
if( checkIfSingular )
{
// TODO: Instead check if values are too small in magnitude
if( gamma11 == F(0) || gamma22 == F(0) )
LogicError("Singular diagonal block detected");
}
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve against Q
const F chi1 = xBuf[k ];
const F chi2 = xBuf[k+1];
xBuf[k ] = c*chi1 + s*chi2;
xBuf[k+1] = -Conj(s)*chi1 + c*chi2;
// Solve against R
xBuf[k+1] /= gamma22;
xBuf[k ] -= gamma12*xBuf[k+1];
xBuf[k ] /= gamma11;
// Update x0 := x0 - U01 x1
blas::Axpy( k, -xBuf[k ], &UBuf[ k *ldu], 1, xBuf, 1 );
blas::Axpy( k, -xBuf[k+1], &UBuf[(k+1)*ldu], 1, xBuf, 1 );
}
}
else
{
if( checkIfSingular )
if( UBuf[k+k*ldu] == F(0) )
LogicError("Singular diagonal entry detected");
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve the 1x1 linear system
xBuf[k] /= UBuf[k+k*ldu];
// Update x0 := x0 - u01 chi_1
blas::Axpy( k, -xBuf[k], &UBuf[k*ldu], 1, xBuf, 1 );
}
}
--k;
}
}
void LLNUnb( const Matrix<F>& L, Matrix<F>& X, bool checkIfSingular )
{
DEBUG_CSE
typedef Base<F> Real;
const Int m = X.Height();
const Int n = X.Width();
const F* LBuf = L.LockedBuffer();
F* XBuf = X.Buffer();
const Int ldl = L.LDim();
const Int ldx = X.LDim();
Int k=0;
while( k < m )
{
const bool in2x2 = ( k+1<m && LBuf[k+(k+1)*ldl] != F(0) );
if( in2x2 )
{
// Solve the 2x2 linear systems via a 2x2 LQ decomposition produced
// by the Givens rotation
// | L(k,k) L(k,k+1) | | c -conj(s) | = | gamma11 0 |
// | s c |
// and by also forming the bottom two entries of the 2x2 resulting
// lower-triangular matrix, say gamma21 and gamma22
//
// Extract the 2x2 diagonal block, D
const F delta11 = LBuf[ k + k *ldl];
const F delta12 = LBuf[ k +(k+1)*ldl];
const F delta21 = LBuf[(k+1)+ k *ldl];
const F delta22 = LBuf[(k+1)+(k+1)*ldl];
// Decompose D = L Q
Real c; F s;
const F gamma11 = Givens( delta11, delta12, c, s );
const F gamma21 = c*delta21 + s*delta22;
const F gamma22 = -Conj(s)*delta21 + c*delta22;
if( checkIfSingular )
{
// TODO: Check if sufficiently small instead
if( gamma11 == F(0) || gamma22 == F(0) )
LogicError("Singular diagonal block detected");
}
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve against L
xBuf[k ] /= gamma11;
xBuf[k+1] -= gamma21*xBuf[k];
xBuf[k+1] /= gamma22;
// Solve against Q
const F chi1 = xBuf[k ];
const F chi2 = xBuf[k+1];
xBuf[k ] = c*chi1 - Conj(s)*chi2;
xBuf[k+1] = s*chi1 + c*chi2;
// Update x2 := x2 - L21 x1
blas::Axpy
( m-(k+2), -xBuf[k ],
&LBuf[(k+2)+ k *ldl], 1, &xBuf[k+2], 1 );
blas::Axpy
( m-(k+2), -xBuf[k+1],
&LBuf[(k+2)+(k+1)*ldl], 1, &xBuf[k+2], 1 );
}
k += 2;
}
else
{
if( checkIfSingular )
{
// TODO: Check if sufficiently small instead
if( LBuf[k+k*ldl] == F(0) )
LogicError("Singular diagonal entry detected");
}
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
xBuf[k] /= LBuf[k+k*ldl];
blas::Axpy
( m-(k+1), -xBuf[k], &LBuf[(k+1)+k*ldl], 1, &xBuf[k+1], 1 );
}
k += 1;
}
}
}
void LLTUnb
( bool conjugate, const Matrix<F>& L, const Matrix<F>& shifts, Matrix<F>& X )
{
DEBUG_CSE
typedef Base<F> Real;
const Int m = X.Height();
const Int n = X.Width();
const F* LBuf = L.LockedBuffer();
F* XBuf = X.Buffer();
const Int ldl = L.LDim();
const Int ldx = X.LDim();
if( conjugate )
Conjugate( X );
Int k=m-1;
while( k >= 0 )
{
const bool in2x2 = ( k>0 && LBuf[(k-1)+k*ldl] != F(0) );
if( in2x2 )
{
--k;
// Solve the 2x2 linear systems via 2x2 LQ decompositions produced
// by the Givens rotation
// | L(k,k)-shift L(k,k+1) | | c -conj(s) | = | gamma11 0 |
// | s c |
// and by also forming the bottom two entries of the 2x2 resulting
// lower-triangular matrix, say gamma21 and gamma22
//
// Extract the constant part of the 2x2 diagonal block, D
const F delta12 = LBuf[ k +(k+1)*ldl];
const F delta21 = LBuf[(k+1)+ k *ldl];
for( Int j=0; j<n; ++j )
{
const F delta11 = LBuf[ k + k *ldl] - shifts.Get(j,0);
const F delta22 = LBuf[(k+1)+(k+1)*ldl] - shifts.Get(j,0);
// Decompose D = L Q
Real c; F s;
const F gamma11 = Givens( delta11, delta12, c, s );
const F gamma21 = c*delta21 + s*delta22;
const F gamma22 = -Conj(s)*delta21 + c*delta22;
F* xBuf = &XBuf[j*ldx];
// Solve against Q^T
const F chi1 = xBuf[k ];
const F chi2 = xBuf[k+1];
xBuf[k ] = c*chi1 + s*chi2;
xBuf[k+1] = -Conj(s)*chi1 + c*chi2;
// Solve against R^T
xBuf[k+1] /= gamma22;
xBuf[k ] -= gamma21*xBuf[k+1];
xBuf[k ] /= gamma11;
// Update x0 := x0 - L10^T x1
blas::Axpy( k, -xBuf[k ], &LBuf[k ], ldl, xBuf, 1 );
blas::Axpy( k, -xBuf[k+1], &LBuf[k+1], ldl, xBuf, 1 );
}
}
else
{
for( Int j=0; j<n; ++j )
{
F* xBuf = &XBuf[j*ldx];
// Solve the 1x1 linear system
xBuf[k] /= LBuf[k+k*ldl] - shifts.Get(j,0);
// Update x0 := x0 - l10^T chi_1
blas::Axpy( k, -xBuf[k], &LBuf[k], ldl, xBuf, 1 );
}
}
--k;
}
if( conjugate )
Conjugate( X );
}