/*************************************************************************
* QR decomposition of a rectangular matrix of size MxN
*
* Input parameters:
* A - matrix A whose indexes range within [1..M, 1..N].
* M - number of rows in matrix A.
* N - number of columns in matrix A.
*
* Output parameters:
* A - matrices Q and R in compact form (see below).
* Tau - array of scalar factors which are used to form
* matrix Q. Array whose index ranges within [1..Min(M,N)].
*
* Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
* MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.
*
* The elements of matrix R are located on and above the main diagonal of
* matrix A. The elements which are located in Tau array and below the main
* diagonal of matrix A are used to form matrix Q as follows:
*
* Matrix Q is represented as a product of elementary reflections
*
* Q = H(1)*H(2)*...*H(k),
*
* where k = min(m,n), and each H(i) is in the form
*
* H(i) = 1 - tau * v * (v^T)
*
* where tau is a scalar stored in Tau[I]; v - real vector,
* so that v(1:i-1) = 0, v(i) = 1, v(i+1:m) stored in A(i+1:m,i).
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*************************************************************************/
public static void Decomposition(ref double[,] a,
int m,
int n,
ref double[] tau)
{
double[] work = new double[0];
double[] t = new double[0];
int i = 0;
int k = 0;
int mmip1 = 0;
int minmn = 0;
double tmp = 0;
int i_ = 0;
int i1_ = 0;
minmn = Math.Min(m, n);
work = new double[n + 1];
t = new double[m + 1];
tau = new double[minmn + 1];
//
// Test the input arguments
//
k = Math.Min(m, n);
for (i = 1; i <= k; i++)
{
//
// Generate elementary reflector H(i) to annihilate A(i+1:m,i)
//
mmip1 = m - i + 1;
i1_ = (i) - (1);
for (i_ = 1; i_ <= mmip1; i_++)
{
t[i_] = a[i_ + i1_, i];
}
Reflections.GenerateReflection(ref t, mmip1, ref tmp);
tau[i] = tmp;
i1_ = (1) - (i);
for (i_ = i; i_ <= m; i_++)
{
a[i_, i] = t[i_ + i1_];
}
t[1] = 1;
if (i < n)
{
//
// Apply H(i) to A(i:m,i+1:n) from the left
//
Reflections.ApplyReflectionFromTheLeft(ref a, tau[i], ref t, i, m, i + 1, n, ref work);
}
}
}
/*************************************************************************
* Multiplication by matrix P which reduces matrix A to bidiagonal form.
*
* The algorithm allows pre- or post-multiply by P or P'.
*
* Input parameters:
* QP - matrices Q and P in compact form.
* Output of ToBidiagonal subroutine.
* M - number of rows in matrix A.
* N - number of columns in matrix A.
* TAUP - scalar factors which are used to form P.
* Output of ToBidiagonal subroutine.
* Z - multiplied matrix.
* Array whose indexes range within [1..ZRows, 1..ZColumns].
* ZRows - number of rows in matrix Z. If FromTheRight=False,
* ZRows=N, otherwise ZRows can be arbitrary.
* ZColumns - number of columns in matrix Z. If FromTheRight=True,
* ZColumns=N, otherwise ZColumns can be arbitrary.
* FromTheRight - pre- or post-multiply.
* DoTranspose - multiply by P or P'.
*
* Output parameters:
* Z - product of Z and P.
* Array whose indexes range within [1..ZRows,1..ZColumns].
* If ZRows=0 or ZColumns=0, the array is not modified.
*
* -- ALGLIB --
* Copyright 2005 by Bochkanov Sergey
*************************************************************************/
public static void MultiplyBypFromBidiagonal(ref double[,] qp,
int m,
int n,
ref double[] taup,
ref double[,] z,
int zrows,
int zcolumns,
bool fromtheright,
bool dotranspose)
{
int i = 0;
int ip1 = 0;
double[] v = new double[0];
double[] work = new double[0];
int vm = 0;
int mx = 0;
int i1 = 0;
int i2 = 0;
int istep = 0;
int i_ = 0;
int i1_ = 0;
if (m <= 0 | n <= 0 | zrows <= 0 | zcolumns <= 0)
{
return;
}
Debug.Assert(fromtheright & zcolumns == n | !fromtheright & zrows == n, "MultiplyByQFromBidiagonal: incorrect Z size!");
//
// init
//
mx = Math.Max(m, n);
mx = Math.Max(mx, zrows);
mx = Math.Max(mx, zcolumns);
v = new double[mx + 1];
work = new double[mx + 1];
v = new double[mx + 1];
work = new double[mx + 1];
if (m >= n)
{
//
// setup
//
if (fromtheright)
{
i1 = n - 1;
i2 = 1;
istep = -1;
}
else
{
i1 = 1;
i2 = n - 1;
istep = +1;
}
if (!dotranspose)
{
i = i1;
i1 = i2;
i2 = i;
istep = -istep;
}
//
// Process
//
if (n - 1 > 0)
{
i = i1;
do
{
vm = n - i;
ip1 = i + 1;
i1_ = (ip1) - (1);
for (i_ = 1; i_ <= vm; i_++)
{
v[i_] = qp[i, i_ + i1_];
}
v[1] = 1;
if (fromtheright)
{
Reflections.ApplyReflectionFromTheRight(ref z, taup[i], ref v, 1, zrows, i + 1, n, ref work);
}
else
{
Reflections.ApplyReflectionFromTheLeft(ref z, taup[i], ref v, i + 1, n, 1, zcolumns, ref work);
}
i = i + istep;
}while (i != i2 + istep);
}
}
else
{
//
// setup
//
if (fromtheright)
{
i1 = m;
i2 = 1;
istep = -1;
}
else
{
i1 = 1;
i2 = m;
istep = +1;
}
if (!dotranspose)
{
i = i1;
i1 = i2;
i2 = i;
istep = -istep;
}
//
// Process
//
i = i1;
do
{
vm = n - i + 1;
i1_ = (i) - (1);
for (i_ = 1; i_ <= vm; i_++)
{
v[i_] = qp[i, i_ + i1_];
}
v[1] = 1;
if (fromtheright)
{
Reflections.ApplyReflectionFromTheRight(ref z, taup[i], ref v, 1, zrows, i, n, ref work);
}
else
{
Reflections.ApplyReflectionFromTheLeft(ref z, taup[i], ref v, i, n, 1, zcolumns, ref work);
}
i = i + istep;
}while (i != i2 + istep);
}
}
/*************************************************************************
* Reduction of a rectangular matrix to bidiagonal form
*
* The algorithm reduces the rectangular matrix A to bidiagonal form by
* orthogonal transformations P and Q: A = Q*B*P.
*
* Input parameters:
* A - source matrix. Array whose indexes range within [1..N, 1..N].
* M - number of rows in matrix A.
* N - number of columns in matrix A.
*
* Output parameters:
* A - matrices Q, B, P in compact form (see below).
* TauQ - scalar factors which are used to form matrix Q.
* TauP - scalar factors which are used to form matrix P.
*
* The main diagonal and one of the secondary diagonals of matrix A are
* replaced with bidiagonal matrix B. Other elements contain elementary
* reflections which form MxM matrix Q and NxN matrix P, respectively.
*
* If M>=N, B is the upper bidiagonal MxN matrix and is stored in the
* corresponding elements of matrix A. Matrix Q is represented as a product
* of elementary reflections Q = H(1)*H(2)*...*H(n), where H(i) = 1 - tau*v*v'.
* Here tau is a scalar which is stored in TauQ[i], and vector v has the
* following structure: v(1:i-1)=0, v(i)=1, v(i+1:m) is stored in elements
* A(i+1:m,i). Matrix P is as follows: P = G(1)*G(2)*...*G(n-1), where
* G(i) = 1 - tau*u*u'. Tau is stored in TauP[i], u(1:i)=0, u(i+1)=1, u(i+2:n)
* is stored in elements A(i,i+2:n).
*
* If M<N, B is the lower bidiagonal MxN matrix and is stored in the
* corresponding elements of matrix A. Q = H(1)*H(2)*...*H(m-1), where
* H(i) = 1 - tau*v*v', tau is stored in TauQ, v(1:i)=0, v(i+1)=1, v(i+2:m)
* is stored in elements A(i+1:m,i). P = G(1)*G(2)*...*G(m), G(i) = 1 - tau*u*u',
* tau is stored in TauP, u(1:i-1)=0, u(i)=1, u(i+1:n) is stored A(i,i+1:n).
*
* EXAMPLE:
*
* m=6, n=5 (m > n): m=5, n=6 (m < n):
*
* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
* ( v1 v2 v3 v4 v5 )
*
* Here vi and ui are vectors which form H(i) and G(i), and d and e -
* are the diagonal and off-diagonal elements of matrix B.
*************************************************************************/
public static void ToBidiagonal(ref double[,] a,
int m,
int n,
ref double[] tauq,
ref double[] taup)
{
double[] work = new double[0];
double[] t = new double[0];
int minmn = 0;
int maxmn = 0;
int i = 0;
double ltau = 0;
int mmip1 = 0;
int nmi = 0;
int ip1 = 0;
int nmip1 = 0;
int mmi = 0;
int i_ = 0;
int i1_ = 0;
minmn = Math.Min(m, n);
maxmn = Math.Max(m, n);
work = new double[maxmn + 1];
t = new double[maxmn + 1];
taup = new double[minmn + 1];
tauq = new double[minmn + 1];
if (m >= n)
{
//
// Reduce to upper bidiagonal form
//
for (i = 1; i <= n; i++)
{
//
// Generate elementary reflector H(i) to annihilate A(i+1:m,i)
//
mmip1 = m - i + 1;
i1_ = (i) - (1);
for (i_ = 1; i_ <= mmip1; i_++)
{
t[i_] = a[i_ + i1_, i];
}
Reflections.GenerateReflection(ref t, mmip1, ref ltau);
tauq[i] = ltau;
i1_ = (1) - (i);
for (i_ = i; i_ <= m; i_++)
{
a[i_, i] = t[i_ + i1_];
}
t[1] = 1;
//
// Apply H(i) to A(i:m,i+1:n) from the left
//
Reflections.ApplyReflectionFromTheLeft(ref a, ltau, ref t, i, m, i + 1, n, ref work);
if (i < n)
{
//
// Generate elementary reflector G(i) to annihilate
// A(i,i+2:n)
//
nmi = n - i;
ip1 = i + 1;
i1_ = (ip1) - (1);
for (i_ = 1; i_ <= nmi; i_++)
{
t[i_] = a[i, i_ + i1_];
}
Reflections.GenerateReflection(ref t, nmi, ref ltau);
taup[i] = ltau;
i1_ = (1) - (ip1);
for (i_ = ip1; i_ <= n; i_++)
{
a[i, i_] = t[i_ + i1_];
}
t[1] = 1;
//
// Apply G(i) to A(i+1:m,i+1:n) from the right
//
Reflections.ApplyReflectionFromTheRight(ref a, ltau, ref t, i + 1, m, i + 1, n, ref work);
}
else
{
taup[i] = 0;
}
}
}
else
{
//
// Reduce to lower bidiagonal form
//
for (i = 1; i <= m; i++)
{
//
// Generate elementary reflector G(i) to annihilate A(i,i+1:n)
//
nmip1 = n - i + 1;
i1_ = (i) - (1);
for (i_ = 1; i_ <= nmip1; i_++)
{
t[i_] = a[i, i_ + i1_];
}
Reflections.GenerateReflection(ref t, nmip1, ref ltau);
taup[i] = ltau;
i1_ = (1) - (i);
for (i_ = i; i_ <= n; i_++)
{
a[i, i_] = t[i_ + i1_];
}
t[1] = 1;
//
// Apply G(i) to A(i+1:m,i:n) from the right
//
Reflections.ApplyReflectionFromTheRight(ref a, ltau, ref t, i + 1, m, i, n, ref work);
if (i < m)
{
//
// Generate elementary reflector H(i) to annihilate
// A(i+2:m,i)
//
mmi = m - i;
ip1 = i + 1;
i1_ = (ip1) - (1);
for (i_ = 1; i_ <= mmi; i_++)
{
t[i_] = a[i_ + i1_, i];
}
Reflections.GenerateReflection(ref t, mmi, ref ltau);
tauq[i] = ltau;
i1_ = (1) - (ip1);
for (i_ = ip1; i_ <= m; i_++)
{
a[i_, i] = t[i_ + i1_];
}
t[1] = 1;
//
// Apply H(i) to A(i+1:m,i+1:n) from the left
//
Reflections.ApplyReflectionFromTheLeft(ref a, ltau, ref t, i + 1, m, i + 1, n, ref work);
}
else
{
tauq[i] = 0;
}
}
}
}
/*************************************************************************
* Unpacking matrix Q which reduces a matrix to bidiagonal form.
*
* Input parameters:
* QP - matrices Q and P in compact form.
* Output of ToBidiagonal subroutine.
* M - number of rows in matrix A.
* N - number of columns in matrix A.
* TAUQ - scalar factors which are used to form Q.
* Output of ToBidiagonal subroutine.
* QColumns - required number of columns in matrix Q.
* M>=QColumns>=0.
*
* Output parameters:
* Q - first QColumns columns of matrix Q.
* Array whose indexes range within [1..M, 1..QColumns].
* If QColumns=0, the array is not modified.
*
* -- ALGLIB --
* Copyright 2005 by Bochkanov Sergey
*************************************************************************/
public static void UnpackqFromBidiagonal(ref double[,] qp,
int m,
int n,
ref double[] tauq,
int qcolumns,
ref double[,] q)
{
int i = 0;
int j = 0;
int ip1 = 0;
double[] v = new double[0];
double[] work = new double[0];
int vm = 0;
int i_ = 0;
int i1_ = 0;
Debug.Assert(qcolumns <= m, "UnpackQFromBidiagonal: QColumns>M!");
if (m == 0 | n == 0 | qcolumns == 0)
{
return;
}
//
// init
//
q = new double[m + 1, qcolumns + 1];
v = new double[m + 1];
work = new double[qcolumns + 1];
//
// prepare Q
//
for (i = 1; i <= m; i++)
{
for (j = 1; j <= qcolumns; j++)
{
if (i == j)
{
q[i, j] = 1;
}
else
{
q[i, j] = 0;
}
}
}
if (m >= n)
{
for (i = Math.Min(n, qcolumns); i >= 1; i--)
{
vm = m - i + 1;
i1_ = (i) - (1);
for (i_ = 1; i_ <= vm; i_++)
{
v[i_] = qp[i_ + i1_, i];
}
v[1] = 1;
Reflections.ApplyReflectionFromTheLeft(ref q, tauq[i], ref v, i, m, 1, qcolumns, ref work);
}
}
else
{
for (i = Math.Min(m - 1, qcolumns - 1); i >= 1; i--)
{
vm = m - i;
ip1 = i + 1;
i1_ = (ip1) - (1);
for (i_ = 1; i_ <= vm; i_++)
{
v[i_] = qp[i_ + i1_, i];
}
v[1] = 1;
Reflections.ApplyReflectionFromTheLeft(ref q, tauq[i], ref v, i + 1, m, 1, qcolumns, ref work);
}
}
}
/*************************************************************************
* Partial unpacking of matrix Q from the QR decomposition of a matrix A
*
* Input parameters:
* A - matrices Q and R in compact form.
* Output of QRDecomposition subroutine.
* M - number of rows in given matrix A. M>=0.
* N - number of columns in given matrix A. N>=0.
* Tau - scalar factors which are used to form Q.
* Output of the QRDecomposition subroutine.
* QColumns - required number of columns of matrix Q. M>=QColumns>=0.
*
* Output parameters:
* Q - first QColumns columns of matrix Q.
* Array whose indexes range within [1..M, 1..QColumns].
* If QColumns=0, the array remains unchanged.
*
* -- ALGLIB --
* Copyright 2005 by Bochkanov Sergey
*************************************************************************/
public static void UnpackqFromQR(ref double[,] a,
int m,
int n,
ref double[] tau,
int qcolumns,
ref double[,] q)
{
int i = 0;
int j = 0;
int k = 0;
int minmn = 0;
double[] v = new double[0];
double[] work = new double[0];
int vm = 0;
int i_ = 0;
int i1_ = 0;
System.Diagnostics.Debug.Assert(qcolumns <= m, "UnpackQFromQR: QColumns>M!");
if (m == 0 | n == 0 | qcolumns == 0)
{
return;
}
//
// init
//
minmn = Math.Min(m, n);
k = Math.Min(minmn, qcolumns);
q = new double[m + 1, qcolumns + 1];
v = new double[m + 1];
work = new double[qcolumns + 1];
for (i = 1; i <= m; i++)
{
for (j = 1; j <= qcolumns; j++)
{
if (i == j)
{
q[i, j] = 1;
}
else
{
q[i, j] = 0;
}
}
}
//
// unpack Q
//
for (i = k; i >= 1; i--)
{
//
// Apply H(i)
//
vm = m - i + 1;
i1_ = (i) - (1);
for (i_ = 1; i_ <= vm; i_++)
{
v[i_] = a[i_ + i1_, i];
}
v[1] = 1;
Reflections.ApplyReflectionFromTheLeft(ref q, tau[i], ref v, i, m, 1, qcolumns, ref work);
}
}