qrfactor.c

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#include <../../nrnconf.h>
 
/**************************************************************************
**
** Copyright (C) 1993 David E. Steward & Zbigniew Leyk, all rights reserved.
**
**               Meschach Library
** 
** This Meschach Library is provided "as is" without any express 
** or implied warranty of any kind with respect to this software. 
** In particular the authors shall not be liable for any direct, 
** indirect, special, incidental or consequential damages arising 
** in any way from use of the software.
** 
** Everyone is granted permission to copy, modify and redistribute this
** Meschach Library, provided:
**  1.  All copies contain this copyright notice.
**  2.  All modified copies shall carry a notice stating who
**      made the last modification and the date of such modification.
**  3.  No charge is made for this software or works derived from it.  
**      This clause shall not be construed as constraining other software
**      distributed on the same medium as this software, nor is a
**      distribution fee considered a charge.
**
***************************************************************************/
 
 
/*
  This file contains the routines needed to perform QR factorisation
  of matrices, as well as Householder transformations.
  The internal "factored form" of a matrix A is not quite standard.
  The diagonal of A is replaced by the diagonal of R -- not by the 1st non-zero
  entries of the Householder vectors. The 1st non-zero entries are held in
  the diag parameter of QRfactor(). The reason for this non-standard
  representation is that it enables direct use of the Usolve() function
  rather than requiring that  a seperate function be written just for this case.
  See, e.g., QRsolve() below for more details.
  
*/
 
 
static  char    rcsid[] = "qrfactor.c,v 1.1 1997/12/04 17:55:45 hines Exp";
 
#include    <stdio.h>
#include        "matrix2.h"
#include    <math.h>
 
 
 
 
 
#define     sign(x) ((x) > 0.0 ? 1 : ((x) < 0.0 ? -1 : 0 ))
 
extern  VEC *Usolve();  /* See matrix2.h */
 
/* Note: The usual representation of a Householder transformation is taken
   to be:
   P = I - beta.u.uT
   where beta = 2/(uT.u) and u is called the Householder vector
   */
 
/* QRfactor -- forms the QR factorisation of A -- factorisation stored in
   compact form as described above ( not quite standard format ) */
/* MAT  *QRfactor(A,diag,beta) */
MAT *QRfactor(A,diag)
MAT *A;
VEC *diag /* ,*beta */;
{
    u_int   k,limit;
    Real    beta;
    static  VEC *tmp1=VNULL;
    
    if ( ! A || ! diag )
    error(E_NULL,"QRfactor");
    limit = min(A->m,A->n);
    if ( diag->dim < limit )
    error(E_SIZES,"QRfactor");
    
    tmp1 = v_resize(tmp1,A->m);
    MEM_STAT_REG(tmp1,TYPE_VEC);
    
    for ( k=0; k<limit; k++ )
    {
    /* get H/holder vector for the k-th column */
    get_col(A,k,tmp1);
    /* hhvec(tmp1,k,&beta->ve[k],tmp1,&A->me[k][k]); */
    hhvec(tmp1,k,&beta,tmp1,&A->me[k][k]);
    diag->ve[k] = tmp1->ve[k];
    
    /* apply H/holder vector to remaining columns */
    /* hhtrcols(A,k,k+1,tmp1,beta->ve[k]); */
    hhtrcols(A,k,k+1,tmp1,beta);
    }
 
    return (A);
}
 
/* QRCPfactor -- forms the QR factorisation of A with column pivoting
   -- factorisation stored in compact form as described above
   ( not quite standard format )                */
/* MAT  *QRCPfactor(A,diag,beta,px) */
MAT *QRCPfactor(A,diag,px)
MAT *A;
VEC *diag /* , *beta */;
PERM    *px;
{
    u_int   i, i_max, j, k, limit;
    static  VEC *gamma=VNULL, *tmp1=VNULL, *tmp2=VNULL;
    Real    beta, maxgamma, sum, tmp;
    
    if ( ! A || ! diag || ! px )
    error(E_NULL,"QRCPfactor");
    limit = min(A->m,A->n);
    if ( diag->dim < limit || px->size != A->n )
    error(E_SIZES,"QRCPfactor");
    
    tmp1 = v_resize(tmp1,A->m);
    tmp2 = v_resize(tmp2,A->m);
    gamma = v_resize(gamma,A->n);
    MEM_STAT_REG(tmp1,TYPE_VEC);
    MEM_STAT_REG(tmp2,TYPE_VEC);
    MEM_STAT_REG(gamma,TYPE_VEC);
    
    /* initialise gamma and px */
    for ( j=0; j<A->n; j++ )
    {
    px->pe[j] = j;
    sum = 0.0;
    for ( i=0; i<A->m; i++ )
        sum += square(A->me[i][j]);
    gamma->ve[j] = sum;
    }
    
    for ( k=0; k<limit; k++ )
    {
    /* find "best" column to use */
    i_max = k;  maxgamma = gamma->ve[k];
    for ( i=k+1; i<A->n; i++ )
        /* Loop invariant:maxgamma=gamma[i_max]
           >=gamma[l];l=k,...,i-1 */
        if ( gamma->ve[i] > maxgamma )
        {   maxgamma = gamma->ve[i]; i_max = i; }
    
    /* swap columns if necessary */
    if ( i_max != k )
    {
        /* swap gamma values */
        tmp = gamma->ve[k];
        gamma->ve[k] = gamma->ve[i_max];
        gamma->ve[i_max] = tmp;
        
        /* update column permutation */
        px_transp(px,k,i_max);
        
        /* swap columns of A */
        for ( i=0; i<A->m; i++ )
        {
        tmp = A->me[i][k];
        A->me[i][k] = A->me[i][i_max];
        A->me[i][i_max] = tmp;
        }
    }
    
    /* get H/holder vector for the k-th column */
    get_col(A,k,tmp1);
    /* hhvec(tmp1,k,&beta->ve[k],tmp1,&A->me[k][k]); */
    hhvec(tmp1,k,&beta,tmp1,&A->me[k][k]);
    diag->ve[k] = tmp1->ve[k];
    
    /* apply H/holder vector to remaining columns */
    /* hhtrcols(A,k,k+1,tmp1,beta->ve[k]); */
    hhtrcols(A,k,k+1,tmp1,beta);
    
    /* update gamma values */
    for ( j=k+1; j<A->n; j++ )
        gamma->ve[j] -= square(A->me[k][j]);
    }
 
    return (A);
}
 
/* Qsolve -- solves Qx = b, Q is an orthogonal matrix stored in compact
   form a la QRfactor() -- may be in-situ */
/* VEC  *_Qsolve(QR,diag,beta,b,x,tmp) */
VEC *_Qsolve(QR,diag,b,x,tmp)
MAT *QR;
VEC *diag /* ,*beta */ , *b, *x, *tmp;
{
    u_int   dynamic;
    int     k, limit;
    Real    beta, r_ii, tmp_val;
    
    limit = min(QR->m,QR->n);
    dynamic = FALSE;
    if ( ! QR || ! diag || ! b )
    error(E_NULL,"_Qsolve");
    if ( diag->dim < limit || b->dim != QR->m )
    error(E_SIZES,"_Qsolve");
    x = v_resize(x,QR->m);
    if ( tmp == VNULL )
    dynamic = TRUE;
    tmp = v_resize(tmp,QR->m);
    
    /* apply H/holder transforms in normal order */
    x = v_copy(b,x);
    for ( k = 0 ; k < limit ; k++ )
    {
    get_col(QR,k,tmp);
    r_ii = fabs(tmp->ve[k]);
    tmp->ve[k] = diag->ve[k];
    tmp_val = (r_ii*fabs(diag->ve[k]));
    beta = ( tmp_val == 0.0 ) ? 0.0 : 1.0/tmp_val;
    /* hhtrvec(tmp,beta->ve[k],k,x,x); */
    hhtrvec(tmp,beta,k,x,x);
    }
    
    if ( dynamic )
    V_FREE(tmp);
    
    return (x);
}
 
/* makeQ -- constructs orthogonal matrix from Householder vectors stored in
   compact QR form */
/* MAT  *makeQ(QR,diag,beta,Qout) */
MAT *makeQ(QR,diag,Qout)
MAT *QR,*Qout;
VEC *diag /* , *beta */;
{
    static  VEC *tmp1=VNULL,*tmp2=VNULL;
    u_int   i, limit;
    Real    beta, r_ii, tmp_val;
    int j;
    
    limit = min(QR->m,QR->n);
    if ( ! QR || ! diag )
    error(E_NULL,"makeQ");
    if ( diag->dim < limit )
    error(E_SIZES,"makeQ");
    if ( Qout==(MAT *)NULL || Qout->m < QR->m || Qout->n < QR->m )
    Qout = m_get(QR->m,QR->m);
    
    tmp1 = v_resize(tmp1,QR->m);    /* contains basis vec & columns of Q */
    tmp2 = v_resize(tmp2,QR->m);    /* contains H/holder vectors */
    MEM_STAT_REG(tmp1,TYPE_VEC);
    MEM_STAT_REG(tmp2,TYPE_VEC);
    
    for ( i=0; i<QR->m ; i++ )
    {   /* get i-th column of Q */
    /* set up tmp1 as i-th basis vector */
    for ( j=0; j<QR->m ; j++ )
        tmp1->ve[j] = 0.0;
    tmp1->ve[i] = 1.0;
    
    /* apply H/h transforms in reverse order */
    for ( j=limit-1; j>=0; j-- )
    {
        get_col(QR,j,tmp2);
        r_ii = fabs(tmp2->ve[j]);
        tmp2->ve[j] = diag->ve[j];
        tmp_val = (r_ii*fabs(diag->ve[j]));
        beta = ( tmp_val == 0.0 ) ? 0.0 : 1.0/tmp_val;
        /* hhtrvec(tmp2,beta->ve[j],j,tmp1,tmp1); */
        hhtrvec(tmp2,beta,j,tmp1,tmp1);
    }
    
    /* insert into Q */
    set_col(Qout,i,tmp1);
    }
 
    return (Qout);
}
 
/* makeR -- constructs upper triangular matrix from QR (compact form)
   -- may be in-situ (all it does is zero the lower 1/2) */
MAT *makeR(QR,Rout)
MAT *QR,*Rout;
{
    u_int   i,j;
    
    if ( QR==(MAT *)NULL )
    error(E_NULL,"makeR");
    Rout = m_copy(QR,Rout);
    
    for ( i=1; i<QR->m; i++ )
    for ( j=0; j<QR->n && j<i; j++ )
        Rout->me[i][j] = 0.0;
    
    return (Rout);
}
 
/* QRsolve -- solves the system Q.R.x=b where Q & R are stored in compact form
   -- returns x, which is created if necessary */
/* VEC  *QRsolve(QR,diag,beta,b,x) */
VEC *QRsolve(QR,diag,b,x)
MAT *QR;
VEC *diag /* , *beta */ , *b, *x;
{
    int limit;
    static  VEC *tmp = VNULL;
    
    if ( ! QR || ! diag || ! b )
    error(E_NULL,"QRsolve");
    limit = min(QR->m,QR->n);
    if ( diag->dim < limit || b->dim != QR->m )
    error(E_SIZES,"QRsolve");
    tmp = v_resize(tmp,limit);
    MEM_STAT_REG(tmp,TYPE_VEC);
 
    x = v_resize(x,QR->n);
    _Qsolve(QR,diag,b,x,tmp);
    x = Usolve(QR,x,x,0.0);
    v_resize(x,QR->n);
 
    return x;
}
 
/* QRCPsolve -- solves A.x = b where A is factored by QRCPfactor()
   -- assumes that A is in the compact factored form */
/* VEC  *QRCPsolve(QR,diag,beta,pivot,b,x) */
VEC *QRCPsolve(QR,diag,pivot,b,x)
MAT *QR;
VEC *diag /* , *beta */;
PERM    *pivot;
VEC *b, *x;
{
    static  VEC *tmp=VNULL;
    
    if ( ! QR || ! diag || ! pivot || ! b )
    error(E_NULL,"QRCPsolve");
    if ( (QR->m > diag->dim &&QR->n > diag->dim) || QR->n != pivot->size )
    error(E_SIZES,"QRCPsolve");
    
    tmp = QRsolve(QR,diag /* , beta */ ,b,tmp);
    MEM_STAT_REG(tmp,TYPE_VEC);
    x = pxinv_vec(pivot,tmp,x);
 
    return x;
}
 
/* Umlt -- compute out = upper_triang(U).x
    -- may be in situ */
static  VEC *Umlt(U,x,out)
MAT *U;
VEC *x, *out;
{
    int     i, limit;
 
    if ( U == MNULL || x == VNULL )
    error(E_NULL,"Umlt");
    limit = min(U->m,U->n);
    if ( limit != x->dim )
    error(E_SIZES,"Umlt");
    if ( out == VNULL || out->dim < limit )
    out = v_resize(out,limit);
 
    for ( i = 0; i < limit; i++ )
    out->ve[i] = __ip__(&(x->ve[i]),&(U->me[i][i]),limit - i);
    return out;
}
 
/* UTmlt -- returns out = upper_triang(U)^T.x */
static  VEC *UTmlt(U,x,out)
MAT *U;
VEC *x, *out;
{
    Real    sum;
    int     i, j, limit;
 
    if ( U == MNULL || x == VNULL )
    error(E_NULL,"UTmlt");
    limit = min(U->m,U->n);
    if ( out == VNULL || out->dim < limit )
    out = v_resize(out,limit);
 
    for ( i = limit-1; i >= 0; i-- )
    {
    sum = 0.0;
    for ( j = 0; j <= i; j++ )
        sum += U->me[j][i]*x->ve[j];
    out->ve[i] = sum;
    }
    return out;
}
 
/* QRTsolve -- solve A^T.sc = c where the QR factors of A are stored in
    compact form
    -- returns sc
    -- original due to Mike Osborne modified Wed 09th Dec 1992 */
VEC *QRTsolve(A,diag,c,sc)
MAT *A;
VEC *diag, *c, *sc;
{
    int     i, j, k, n, p;
    Real    beta, r_ii, s, tmp_val;
 
    if ( ! A || ! diag || ! c )
    error(E_NULL,"QRTsolve");
    if ( diag->dim < min(A->m,A->n) )
    error(E_SIZES,"QRTsolve");
    sc = v_resize(sc,A->m);
    n = sc->dim;
    p = c->dim;
    if ( n == p )
    k = p-2;
    else
    k = p-1;
    v_zero(sc);
    sc->ve[0] = c->ve[0]/A->me[0][0];
    if ( n ==  1)
    return sc;
    if ( p > 1)
    {
    for ( i = 1; i < p; i++ )
    {
        s = 0.0;
        for ( j = 0; j < i; j++ )
        s += A->me[j][i]*sc->ve[j];
        if ( A->me[i][i] == 0.0 )
        error(E_SING,"QRTsolve");
        sc->ve[i]=(c->ve[i]-s)/A->me[i][i];
    }
    }
    for (i = k; i >= 0; i--)
    {
    s = diag->ve[i]*sc->ve[i];
    for ( j = i+1; j < n; j++ )
        s += A->me[j][i]*sc->ve[j];
    r_ii = fabs(A->me[i][i]);
    tmp_val = (r_ii*fabs(diag->ve[i]));
    beta = ( tmp_val == 0.0 ) ? 0.0 : 1.0/tmp_val;
    tmp_val = beta*s;
    sc->ve[i] -= tmp_val*diag->ve[i];
    for ( j = i+1; j < n; j++ )
        sc->ve[j] -= tmp_val*A->me[j][i];
    }
 
    return sc;
}
 
/* QRcondest -- returns an estimate of the 2-norm condition number of the
        matrix factorised by QRfactor() or QRCPfactor()
    -- note that as Q does not affect the 2-norm condition number,
        it is not necessary to pass the diag, beta (or pivot) vectors
    -- generates a lower bound on the true condition number
    -- if the matrix is exactly singular, HUGE is returned
    -- note that QRcondest() is likely to be more reliable for
        matrices factored using QRCPfactor() */
double  QRcondest(QR)
MAT *QR;
{
    static  VEC *y=VNULL;
    Real    norm1, norm2, sum, tmp1, tmp2;
    int     i, j, limit;
 
    if ( QR == MNULL )
    error(E_NULL,"QRcondest");
 
    limit = min(QR->m,QR->n);
    for ( i = 0; i < limit; i++ )
    if ( QR->me[i][i] == 0.0 )
        return HUGE;
 
    y = v_resize(y,limit);
    MEM_STAT_REG(y,TYPE_VEC);
    /* use the trick for getting a unit vector y with ||R.y||_inf small
       from the LU condition estimator */
    for ( i = 0; i < limit; i++ )
    {
    sum = 0.0;
    for ( j = 0; j < i; j++ )
        sum -= QR->me[j][i]*y->ve[j];
    sum -= (sum < 0.0) ? 1.0 : -1.0;
    y->ve[i] = sum / QR->me[i][i];
    }
    UTmlt(QR,y,y);
 
    /* now apply inverse power method to R^T.R */
    for ( i = 0; i < 3; i++ )
    {
    tmp1 = v_norm2(y);
    sv_mlt(1/tmp1,y,y);
    UTsolve(QR,y,y,0.0);
    tmp2 = v_norm2(y);
    sv_mlt(1/v_norm2(y),y,y);
    Usolve(QR,y,y,0.0);
    }
    /* now compute approximation for ||R^{-1}||_2 */
    norm1 = sqrt(tmp1)*sqrt(tmp2);
 
    /* now use complementary approach to compute approximation to ||R||_2 */
    for ( i = limit-1; i >= 0; i-- )
    {
    sum = 0.0;
    for ( j = i+1; j < limit; j++ )
        sum += QR->me[i][j]*y->ve[j];
    y->ve[i] = (sum >= 0.0) ? 1.0 : -1.0;
    y->ve[i] = (QR->me[i][i] >= 0.0) ? y->ve[i] : - y->ve[i];
    }
 
    /* now apply power method to R^T.R */
    for ( i = 0; i < 3; i++ )
    {
    tmp1 = v_norm2(y);
    sv_mlt(1/tmp1,y,y);
    Umlt(QR,y,y);
    tmp2 = v_norm2(y);
    sv_mlt(1/tmp2,y,y);
    UTmlt(QR,y,y);
    }
    norm2 = sqrt(tmp1)*sqrt(tmp2);
 
    /* printf("QRcondest: norm1 = %g, norm2 = %g\n",norm1,norm2); */
 
    return norm1*norm2;
}