klu2_diagnostics.hpp

/* ========================================================================== */
/* === KLU_diagnostics ====================================================== */
/* ========================================================================== */
// @HEADER
// *****************************************************************************
//                   KLU2: A Direct Linear Solver package
//
// Copyright 2011 NTESS and the KLU2 contributors.
// SPDX-License-Identifier: LGPL-2.1-or-later
// *****************************************************************************
// @HEADER
 
/* Linear algebraic diagnostics:
 * KLU_rgrowth: reciprocal pivot growth, takes O(|A|+|U|) time
 * KLU_condest: condition number estimator, takes about O(|A|+5*(|L|+|U|)) time
 * KLU_flops:   compute # flops required to factorize A into L*U
 * KLU_rcond:   compute a really cheap estimate of the reciprocal of the
 *              condition number, min(abs(diag(U))) / max(abs(diag(U))).
 *              Takes O(n) time.
 */
 
#ifndef KLU2_DIAGNOSTICS_HPP
#define KLU2_DIAGNOSTICS_HPP
 
#include "klu2_internal.h"
#include "klu2_tsolve.hpp"
 
/* ========================================================================== */
/* === KLU_rgrowth ========================================================== */
/* ========================================================================== */
 
/* Compute the reciprocal pivot growth factor.  In MATLAB notation:
 *
 *   rgrowth = min (max (abs ((R \ A (p,q)) - F))) ./ max (abs (U)))
 */
 
template <typename Entry, typename Int>
Int KLU_rgrowth         /* return TRUE if successful, FALSE otherwise */
(
    Int *Ap,
    Int *Ai,
    double *Ax,
    KLU_symbolic<Entry, Int> *Symbolic,
    KLU_numeric<Entry, Int> *Numeric,
    KLU_common<Entry, Int> *Common
)
{
    double temp, max_ai, max_ui, min_block_rgrowth ;
    Entry aik ;
    Int *Q, *Ui, *Uip, *Ulen, *Pinv ;
    Unit *LU ;
    Entry *Aentry, *Ux, *Ukk ;
    double *Rs ;
    Int i, newrow, oldrow, k1, k2, nk, j, oldcol, k, pend, len ;
 
    /* ---------------------------------------------------------------------- */
    /* check inputs */
    /* ---------------------------------------------------------------------- */
 
    if (Common == NULL)
    {
        return (FALSE) ;
    }
 
    if (Symbolic == NULL || Ap == NULL || Ai == NULL || Ax == NULL)
    {
        Common->status = KLU_INVALID ;
        return (FALSE) ;
    }
 
    if (Numeric == NULL)
    {
        /* treat this as a singular matrix */
        Common->rgrowth = 0 ;
        Common->status = KLU_SINGULAR ;
        return (TRUE) ;
    }
    Common->status = KLU_OK ;
 
    /* ---------------------------------------------------------------------- */
    /* compute the reciprocal pivot growth */
    /* ---------------------------------------------------------------------- */
 
    Aentry = (Entry *) Ax ;
    Pinv = Numeric->Pinv ;
    Rs = Numeric->Rs ;
    Q = Symbolic->Q ;
    Common->rgrowth = 1 ;
 
    for (i = 0 ; i < Symbolic->nblocks ; i++)
    {
        k1 = Symbolic->R[i] ;
        k2 = Symbolic->R[i+1] ;
        nk = k2 - k1 ;
        if (nk == 1)
        {
            continue ;      /* skip singleton blocks */
        }
        LU = (Unit *) Numeric->LUbx[i] ;
        Uip = Numeric->Uip + k1 ;
        Ulen = Numeric->Ulen + k1 ;
        Ukk = ((Entry *) Numeric->Udiag) + k1 ;
        min_block_rgrowth = 1 ;
        for (j = 0 ; j < nk ; j++)
        {
            max_ai = 0 ;
            max_ui = 0 ;
            oldcol = Q[j + k1] ;
            pend = Ap [oldcol + 1] ;
            for (k = Ap [oldcol] ; k < pend ; k++)
            {
                oldrow = Ai [k] ;
                newrow = Pinv [oldrow] ;
                if (newrow < k1)
                {
                    continue ;  /* skip entry outside the block */
                }
                ASSERT (newrow < k2) ;
                if (Rs != NULL)
                {
                    /* aik = Aentry [k] / Rs [oldrow] */
                    SCALE_DIV_ASSIGN (aik, Aentry [k], Rs [newrow]) ;
                }
                else
                {
                    aik = Aentry [k] ;
                }
                /* temp = ABS (aik) */
                KLU2_ABS (temp, aik) ;
                if (temp > max_ai)
                {
                    max_ai = temp ;
                }
            }
 
            GET_POINTER (LU, Uip, Ulen, Ui, Ux, j, len) ;
            for (k = 0 ; k < len ; k++)
            {
                /* temp = ABS (Ux [k]) */
                KLU2_ABS (temp, Ux [k]) ;
                if (temp > max_ui)
                {
                    max_ui = temp ;
                }
            }
            /* consider the diagonal element */
            KLU2_ABS (temp, Ukk [j]) ;
            if (temp > max_ui)
            {
                max_ui = temp ;
            }
 
            /* if max_ui is 0, skip the column */
            if (SCALAR_IS_ZERO (max_ui))
            {
                continue ;
            }
            temp = max_ai / max_ui ;
            if (temp < min_block_rgrowth)
            {
                min_block_rgrowth = temp ;
            }
        }
 
        if (min_block_rgrowth < Common->rgrowth)
        {
            Common->rgrowth = min_block_rgrowth ;
        }
    }
    return (TRUE) ;
}
 
 
/* ========================================================================== */
/* === KLU_condest ========================================================== */
/* ========================================================================== */
 
/* Estimate the condition number.  Uses Higham and Tisseur's algorithm
 * (A block algorithm for matrix 1-norm estimation, with applications to
 * 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., 21(4):1185-1201, 2000.
 */
 
template <typename Entry, typename Int>
Int KLU_condest         /* return TRUE if successful, FALSE otherwise */
(
    Int Ap [ ],
    double Ax [ ], /* TODO : Check !!! */
    KLU_symbolic<Entry, Int> *Symbolic,
    KLU_numeric<Entry, Int> *Numeric,
    KLU_common<Entry, Int> *Common
)
{
    double xj, Xmax, csum, anorm, ainv_norm, est_old, est_new, abs_value ;
    Entry *Udiag, *Aentry, *X, *S ;
    Int *R ;
    Int nblocks, i, j, jmax, jnew, pend, n ;
#ifndef COMPLEX
    Int unchanged ;
#endif
 
    /* ---------------------------------------------------------------------- */
    /* check inputs */
    /* ---------------------------------------------------------------------- */
 
    if (Common == NULL)
    {
        return (FALSE) ;
    }
    if (Symbolic == NULL || Ap == NULL || Ax == NULL)
    {
        Common->status = KLU_INVALID ;
        return (FALSE) ;
    }
    abs_value = 0 ;
    if (Numeric == NULL)
    {
        /* treat this as a singular matrix */
        Common->condest = 1 / abs_value ;
        Common->status = KLU_SINGULAR ;
        return (TRUE) ;
    }
    Common->status = KLU_OK ;
 
    /* ---------------------------------------------------------------------- */
    /* get inputs */
    /* ---------------------------------------------------------------------- */
 
    n = Symbolic->n ;
    nblocks = Symbolic->nblocks ;
    R = Symbolic->R ;
    Udiag = (Entry *) Numeric->Udiag ;
 
    /* ---------------------------------------------------------------------- */
    /* check if diagonal of U has a zero on it */
    /* ---------------------------------------------------------------------- */
 
    for (i = 0 ; i < n ; i++)
    {
        KLU2_ABS (abs_value, Udiag [i]) ;
        if (SCALAR_IS_ZERO (abs_value))
        {
            Common->condest = 1 / abs_value ;
            Common->status = KLU_SINGULAR ;
            return (TRUE) ;
        }
    }
 
    /* ---------------------------------------------------------------------- */
    /* compute 1-norm (maximum column sum) of the matrix */
    /* ---------------------------------------------------------------------- */
 
    anorm =  0.0 ;
    Aentry = (Entry *) Ax ;
    for (i = 0 ; i < n ; i++)
    {
        pend = Ap [i + 1] ;
        csum = 0.0 ;
        for (j = Ap [i] ; j < pend ; j++)
        {
            KLU2_ABS (abs_value, Aentry [j]) ;
            csum += abs_value ;
        }
        if (csum > anorm)
        {
            anorm = csum ;
        }
    }
 
    /* ---------------------------------------------------------------------- */
    /* compute estimate of 1-norm of inv (A) */
    /* ---------------------------------------------------------------------- */
 
    /* get workspace (size 2*n Entry's) */
    X = (Entry *) Numeric->Xwork ;  /* size n space used in KLU_solve, tsolve */
    X += n ;                        /* X is size n */
    S = X + n ;                     /* S is size n */
 
    for (i = 0 ; i < n ; i++)
    {
        CLEAR (S [i]) ;
        CLEAR (X [i]) ;
        /* TODO : Check REAL(X[i]) -> X[i]*/
        /*REAL (X [i]) = 1.0 / ((double) n) ;*/
        X [i] = 1.0 / ((double) n) ;
    }
    jmax = 0 ;
 
    ainv_norm = 0.0 ;
    for (i = 0 ; i < 5 ; i++)
    {
        if (i > 0)
        {
            /* X [jmax] is the largest entry in X */
            for (j = 0 ; j < n ; j++)
            {
                /* X [j] = 0 ;*/
                CLEAR (X [j]) ;
            }
            /* TODO : Check REAL(X[i]) -> X[i]*/
            /*REAL (X [jmax]) = 1 ;*/
            X [jmax] = 1 ;
        }
 
        KLU_solve (Symbolic, Numeric, n, 1, (double *) X, Common) ;
        est_old = ainv_norm ;
        ainv_norm = 0.0 ;
 
        for (j = 0 ; j < n ; j++)
        {
            /* ainv_norm += ABS (X [j]) ;*/
            KLU2_ABS (abs_value, X [j]) ;
            ainv_norm += abs_value ;
        }
 
#ifndef COMPLEX
        unchanged = TRUE ;
 
        for (j = 0 ; j < n ; j++)
        {
            double s = (X [j] >= 0) ? 1 : -1 ;
            if (s != (Int) REAL (S [j]))
            {
                S [j] = s ;
                unchanged = FALSE ;
            }
        }
 
        if (i > 0 && (ainv_norm <= est_old || unchanged))
        {
            break ;
        }
#else
        for (j = 0 ; j < n ; j++)
        {
            if (IS_NONZERO (X [j]))
            {
                KLU2_ABS (abs_value, X [j]) ;
                SCALE_DIV_ASSIGN (S [j], X [j], abs_value) ;
            }
            else
            {
                CLEAR (S [j]) ;
                /* TODO : Check REAL(X[i]) -> X[i]*/
                /*REAL (S [j]) = 1 ; */
                S [j] = 1 ;
            }
        }
 
        if (i > 0 && ainv_norm <= est_old)
        {
            break ;
        }
#endif
 
        for (j = 0 ; j < n ; j++)
        {
            X [j] = S [j] ;
        }
 
#ifndef COMPLEX
        /* do a transpose solve */
        KLU_tsolve (Symbolic, Numeric, n, 1, X, Common) ;
#else
        /* do a conjugate transpose solve */
        KLU_tsolve (Symbolic, Numeric, n, 1, (double *) X, 1, Common) ;
#endif
 
        /* jnew = the position of the largest entry in X */
        jnew = 0 ;
        Xmax = 0 ;
        for (j = 0 ; j < n ; j++)
        {
            /* xj = ABS (X [j]) ;*/
            KLU2_ABS (xj, X [j]) ;
            if (xj > Xmax)
            {
                Xmax = xj ;
                jnew = j ;
            }
        }
        if (i > 0 && jnew == jmax)
        {
            /* the position of the largest entry did not change
             * from the previous iteration */
            break ;
        }
        jmax = jnew ;
    }
 
    /* ---------------------------------------------------------------------- */
    /* compute another estimate of norm(inv(A),1), and take the largest one */
    /* ---------------------------------------------------------------------- */
 
    for (j = 0 ; j < n ; j++)
    {
        CLEAR (X [j]) ;
        if (j % 2)
        {
            /* TODO : Check REAL(X[i]) -> X[i]*/
            /*REAL (X [j]) = 1 + ((double) j) / ((double) (n-1)) ;*/
            X [j] = 1 + ((double) j) / ((double) (n-1)) ;
        }
        else
        {
            /* TODO : Check REAL(X[i]) -> X[i]*/
            /*REAL (X [j]) = -1 - ((double) j) / ((double) (n-1)) ;*/
            X [j] = -1 - ((double) j) / ((double) (n-1)) ;
        }
    }
 
    KLU_solve (Symbolic, Numeric, n, 1, (double *) X, Common) ;
 
    est_new = 0.0 ;
    for (j = 0 ; j < n ; j++)
    {
        /* est_new += ABS (X [j]) ;*/
        KLU2_ABS (abs_value, X [j]) ;
        est_new += abs_value ;
    }
    est_new = 2 * est_new / (3 * n) ;
    ainv_norm = MAX (est_new, ainv_norm) ;
 
    /* ---------------------------------------------------------------------- */
    /* compute estimate of condition number */
    /* ---------------------------------------------------------------------- */
 
    Common->condest = ainv_norm * anorm ;
    return (TRUE) ;
}
 
 
/* ========================================================================== */
/* === KLU_flops ============================================================ */
/* ========================================================================== */
 
/* Compute the flop count for the LU factorization (in Common->flops) */
 
template <typename Entry, typename Int>
Int KLU_flops           /* return TRUE if successful, FALSE otherwise */
(
    KLU_symbolic<Entry, Int> *Symbolic,
    KLU_numeric<Entry, Int> *Numeric,
    KLU_common<Entry, Int> *Common
)
{
    double flops = 0 ;
    Int *R, *Ui, *Uip, *Llen, *Ulen ;
    Unit **LUbx ;
    Unit *LU ;
    Int k, ulen, p, n, nk, block, nblocks, k1 ;
 
    /* ---------------------------------------------------------------------- */
    /* check inputs */
    /* ---------------------------------------------------------------------- */
 
    if (Common == NULL)
    {
        return (FALSE) ;
    }
    Common->flops = AMESOS2_KLU2_EMPTY ;
    if (Numeric == NULL || Symbolic == NULL)
    {
        Common->status = KLU_INVALID ;
        return (FALSE) ;
    }
    Common->status = KLU_OK ;
 
    /* ---------------------------------------------------------------------- */
    /* get the contents of the Symbolic object */
    /* ---------------------------------------------------------------------- */
 
    n = Symbolic->n ;
    R = Symbolic->R ;
    nblocks = Symbolic->nblocks ;
 
    /* ---------------------------------------------------------------------- */
    /* get the contents of the Numeric object */
    /* ---------------------------------------------------------------------- */
 
    LUbx = (Unit **) Numeric->LUbx ;
 
    /* ---------------------------------------------------------------------- */
    /* compute the flop count */
    /* ---------------------------------------------------------------------- */
 
    for (block = 0 ; block < nblocks ; block++)
    {
        k1 = R [block] ;
        nk = R [block+1] - k1 ;
        if (nk > 1)
        {
            Llen = Numeric->Llen + k1 ;
            Uip  = Numeric->Uip  + k1 ;
            Ulen = Numeric->Ulen + k1 ;
            LU = LUbx [block] ;
            for (k = 0 ; k < nk ; k++)
            {
                /* compute kth column of U, and update kth column of A */
                GET_I_POINTER (LU, Uip, Ui, k) ;
                ulen = Ulen [k] ;
                for (p = 0 ; p < ulen ; p++)
                {
                    flops += 2 * Llen [Ui [p]] ;
                }
                /* gather and divide by pivot to get kth column of L */
                flops += Llen [k] ;
            }
        }
    }
    Common->flops = flops ;
    return (TRUE) ;
}
 
 
/* ========================================================================== */
/* === KLU_rcond ============================================================ */
/* ========================================================================== */
 
/* Compute a really cheap estimate of the reciprocal of the condition number,
 * condition number, min(abs(diag(U))) / max(abs(diag(U))).  If U has a zero
 * pivot, or a NaN pivot, rcond will be zero.  Takes O(n) time.
 */   
 
template <typename Entry, typename Int>
Int KLU_rcond           /* return TRUE if successful, FALSE otherwise */
(
    KLU_symbolic<Entry, Int> *Symbolic,     /* input, not modified */
    KLU_numeric<Entry, Int> *Numeric,       /* input, not modified */
    KLU_common<Entry, Int> *Common          /* result in Common->rcond */
)
{
    double ukk, umin = 0, umax = 0 ;
    Entry *Udiag ;
    Int j, n ;
 
    /* ---------------------------------------------------------------------- */
    /* check inputs */
    /* ---------------------------------------------------------------------- */
 
    if (Common == NULL)
    {
        return (FALSE) ;
    }
    if (Symbolic == NULL)
    {
        Common->status = KLU_INVALID ;
        return (FALSE) ;
    }
    if (Numeric == NULL)
    {
        Common->rcond = 0 ;
        Common->status = KLU_SINGULAR ;
        return (TRUE) ;
    }
    Common->status = KLU_OK ;
 
    /* ---------------------------------------------------------------------- */
    /* compute rcond */
    /* ---------------------------------------------------------------------- */
 
    n = Symbolic->n ;
    Udiag = (Entry *) Numeric->Udiag ;
    for (j = 0 ; j < n ; j++)
    {
        /* get the magnitude of the pivot */
        KLU2_ABS (ukk, Udiag [j]) ;
        if (SCALAR_IS_NAN (ukk) || SCALAR_IS_ZERO (ukk))
        {
            /* if NaN, or zero, the rcond is zero */
            Common->rcond = 0 ;
            Common->status = KLU_SINGULAR ;
            return (TRUE) ;
        }
        if (j == 0)
        {
            /* first pivot entry */
            umin = ukk ;
            umax = ukk ;
        }
        else
        {
            /* subsequent pivots */
            umin = MIN (umin, ukk) ;
            umax = MAX (umax, ukk) ;
        }
    }
 
    Common->rcond = umin / umax ;
    if (SCALAR_IS_NAN (Common->rcond) || SCALAR_IS_ZERO (Common->rcond))
    {
        /* this can occur if umin or umax are Inf or NaN */
        Common->rcond = 0 ;
        Common->status = KLU_SINGULAR ;
    }
    return (TRUE) ;
}
 
#endif