Dreieckslöser

Für eine untere \(n\times n\) Dreiecksmatrix \(L\) soll mit der Funktion dtrlsv (double triangular lower solve vector) die Operation

\[x \leftarrow L^{-1} x\]

durchgeführt werden. Der Vektor \(x\) soll also mit der Lösung \(L^{-1} x\) überschrieben werden. Bei der Herleitung unterscheiden wir aber formal zwischen dem Vektor vor und nach der Operation und definieren

\[z := L^{-1} x\]

als Lösung des Problems. Dann muss nach entsprechender Partitionierung von \(x\) und \(z\) gelten:

\[\left(\begin{array}{c} x_0 \\ \hline x_k \end{array}\right)=\left(\begin{array}{c|c} L_{0,0} & \\ \hline L_{k,0} & L_{k,k} \end{array}\right)\left(\begin{array}{c} z_0 \\ \hline z_k \end{array}\right)=\left(\begin{array}{c} L_{0,0} z_0 \\ \hline L_{k,0} z_0 + L_{k,k} z_k \end{array}\right)\quad\Longleftrightarrow\quad\left\{\begin{array}{lcl} z_0 &=& L_{0,0}^{-1} \; x_0 \\ L_{k,k} z_k &=& x_k - L_{k,0}\, z_0 \end{array}\right.\]

Wir leiten daraus zwei Varianten her, mit denen die Operation iterativ mit den Zwischenergebnissen

\[x = x^{(0)}\leadsto x^{(1)}\leadsto\dots x^{(n)} = z\]

berechnet wird:

In der ersten Variante (Row-Major) soll für \(x^{(k)}\) gelten:
\[x^{(k)} =\left(\begin{array}{c} x_0^{(k)} \\ \hline x_k^{(k)} \end{array}\right)=\left(\begin{array}{c} L_{0,0}^{-1} \; x_0 \\ \hline x_k\end{array}\right)=\left(\begin{array}{c} z_0 \\ \hline x_k\end{array}\right)\]
In der zweiten Variante (Col-Major) soll für \(x^{(k)}\) gelten:
\[x^{(k)} =\left(\begin{array}{c} x_0^{(k)} \\ \hline x_k^{(k)} \end{array}\right)=\left(\begin{array}{c} L_{0,0}^{-1} x_0 \\ \hline x_k - L_{k,0}\, z_0 \end{array}\right)=\left(\begin{array}{c} z_0 \\ \hline x_k - L_{k,0}\, z_0\end{array}\right)\]

Dass die Bezeichnungen Row-Major und Col-Major berechtigt sind, wird allerdings erst im Nachhinein deutlich.

Row-Major (Unblocked)

Wir betrachten die gemeinsamen Partitionierungen

\[ x^{(k)} = \left(\begin{array}{c} x_0^{(k)} \\ \hline x_k^{(k)} \\ \hdashline x_{k+1}^{(k)} \end{array}\right) \qquad\text{und}\qquad x^{(k+1)} = \left(\begin{array}{c} x_0^{(k+1)} \\ \hdashline x_k^{(k+1)} \\ \hline x_{k+1}^{(k+1)} \end{array}\right)\]

sowie

\[L = \left(\begin{array}{c|c|c} L_{0,0} & & \\ \hline \ell_{k,0}^T & \ell_{k,k} & \\ \hline L_{k+1,0} & \ell_{k+1,k} & L_{k+1,k+1} \\ \end{array}\right)\]

Für die Durchführung einer Iteration rechnen wir nun nach:

Laut Voraussetzung gilt für \(x^{(k)}\):
\[L_{0,0} x_0^{(k)} = x_0\qquad\Longleftrightarrow\qquad x_0^{(k)} = z_0\]
Für \(x^{(k+1)}\) soll nach der Iteration
\[\begin{array}{lclcl}L_{0,0} x_0^{(k+1)} & & &=& x_0 \\\ell_{k,0}^T x_0^{(k+1)} &+& \ell_{k,k} x_k^{(k+1)} &=& x_k\end{array}\]
gelten.

Offensichtlich ist also \(x_0^{(k+1)} = x_0^{(k)} = z_0\) und

\[x_k^{(k+1)} = \frac{x_k - \ell_{k,0}^T x_0^{(k+1)}}{\ell_{k,k}}.\]

Um \(x\) mit \(z\) zu überschreiben, erhalten wir als Verfahren

For \(k=0, \dots, n-1\):
- \(x_k \leftarrow x_k - \ell_{k,0}^T x_0\) (DOT)
- \(x_k \leftarrow \frac{x_k}{\ell_{k,k}}\)

Wie es die Prophezeiung besagte, wird hier nur auf Zeilen von \(L\) zugegriffen.

Col-Major (Unblocked)

Laut Voraussetzung gilt für \(x^{(k)}\):
\[L_{0,0} x_0^{(k)} = x_0\qquad\Longleftrightarrow\qquad x_0^{(k)} = z_0\]
und
\[\left(\begin{array}{c} x_k^{(k)} \\ \hdashline x_{k+1}^{(k)} \end{array}\right)=\left(\begin{array}{c} x_k \\ \hdashline x_{k+1} \end{array}\right)-\left(\begin{array}{c} \ell_{k,0}^T \\ \hdashline L_{k+1,0} \end{array}\right) z_0\qquad\Longleftrightarrow\qquad\left\{\begin{array}{lclcl} x_k^{(k)} &=& x_k &-& \ell_{k,0}^T z_0 \\ x_{k+1}^{(k)} &=& x_{k+1} &-& L_{k+1,0} z_0 \end{array}\right.\]
Für \(x^{(k+1)}\) soll nach der Iteration
\[\left(\begin{array}{c:c} L_{0,0} & \\ \hdashline \ell_{k,0}^T & \ell_{k,k} \\ \end{array}\right)\left(\begin{array}{c} x_0^{(k+1)} \\ \hdashline x_k^{(k+1)} \end{array}\right)=\left(\begin{array}{c} x_0 \\ \hdashline x_k \end{array}\right)\qquad\Longleftrightarrow\qquad\left\{\begin{array}{lclcl} L_{0,0} x_0^{(k+1)} & & &=& x_0 \\ \ell_{k,0}^T x_0^{(k+1)} &+& \ell_{k,k} x_k^{(k+1)} &=& x_k \end{array}\right.\qquad\Longleftrightarrow\qquad\left\{\begin{array}{lcl} x_0^{(k+1)} &=& z_0 \\ \ell_{k,k} x_k^{(k+1)} &=& x_k - \ell_{k,0}^T z_0 \end{array}\right.\]
und
\[\begin{array}{lcl}x_{k+1}^{(k+1)}&=&x_{k+1}-\left(\begin{array}{c:c} L_{k+1,0} & \ell_{k+1,k} \end{array}\right)\left(\begin{array}{c} x_0^{(k+1)}\\ \hdashline x_k^{(k+1)} \end{array}\right)\\&=& x_{k+1} - L_{k+1,0} x_0^{(k+1)} - \ell_{k+1,k} x_k^{(k+1)}\\&=& x_{k+1} - L_{k+1,0} z_0 - \ell_{k+1,k} x_k^{(k+1)}\\&=& x_{k+1}^{(k)} - \ell_{k+1,k} x_k^{(k+1)}\\\end{array}\]
gelten.

Wegen \(x_0^{(k+1)} = x_0^{(k)} = z_0\) muss also nur

\[\begin{array}{lcl}x_k^{(k+1)} &=& \frac{1}{\ell_{k,k}}\left(x_k - \ell_{k,0}^T z_0\right) = \frac{1}{\ell_{k,k}} x_k^{(k)} \\x_{k+1}^{(k+1)} &=& x_{k+1}^{(k)} - \ell_{k+1,k} x_k^{(k+1)}\end{array}\]

berechnet werden. Als Algorithmus erhalten wir somit:

For \(k=0, \dots, n-1\):
- \(x_k \leftarrow \frac{1}{\ell_{k,k}} x_k\)
- \(x_{k+1} \leftarrow x_{k+1} - \ell_{k+1,k} x_k\) (AXPY)

Aufgabe

Implementiert die Prozeduren dtrlsv_row und dtrlsv_col in der untenstehenden Vorlage.

Testet eure Implementierung jeweils für zeilen- und spaltenweise gespeicherte Matrizen. Übersetzt dazu wiederum das Programm jeweils mit -DROWMAJOR=1 und -DROWMAJOR=0.
Verwendet verschiedene Optimierungen: -O1, -O3 und -Ofast.

session06/example01/bench_trlsv.c

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <sys/times.h>
#include <unistd.h>

//-- timer for benchmarks ------------------------------------------------------

double
walltime()
{
   struct tms    ts;
   static double ClockTick=0.0;

   if (ClockTick==0.0) {
        ClockTick = 1.0 / ((double) sysconf(_SC_CLK_TCK));
   }
   return ((double) times(&ts)) * ClockTick;
}

//-- setup and print matrices --------------------------------------------------

void
initGeMatrix(int m, int n, double *A, int incRowA, int incColA)
{
    int i, j;

    for (i=0; i<m; ++i) {
        for (j=0; j<n; ++j) {
            A[i*incRowA+j*incColA] = i*n + j + 1;
        }
    }
}

void
randGeMatrix(int m, int n, double *A, int incRowA, int incColA)
{
    int i, j;
    for (j=0; j<n; ++j) {
        for (i=0; i<m; ++i) {
            A[i*incRowA+j*incColA] = ((double)rand()-RAND_MAX/2)*200/RAND_MAX;
        }
    }
}

void
makeTrlDiagDom(int n, int unitDiag, double *A, int incRowA, int incColA)
{
    int i, j;

    for (j=0; j<n; ++j) {
        double asum = 0;
        double A_jj = (unitDiag) ? 1
                                 : A[j*(incRowA+incColA)];
        for (i=j+1; i<n; ++i) {
            asum += fabs(A[i*incRowA+j*incColA]);
        }
        for (i=j+1; i<n; ++i) {
            A[i*incRowA+j*incColA] *= A_jj/(asum*1.1);
        }
    }
}

void
printGeMatrix(int m, int n, const double *A, int incRowA, int incColA)
{
    int i, j;

    for (i=0; i<m; ++i) {
        for (j=0; j<n; ++j) {
            printf("%10.4lf ", A[i*incRowA+j*incColA]);
        }
        printf("\n");
    }
    printf("\n\n");
}

void
printTrMatrix(int m, int n, int unitDiag, int lower,
              const double *A, int incRowA, int incColA)
{
    int i, j;

    for (i=0; i<m; ++i) {
        for (j=0; j<n; ++j) {
            if (unitDiag && (i==j)) {
                printf("%10.4lf ", 1.0);
            } else if ((lower && (i>=j)) || (!lower && (i<=j))) {
                printf("%10.4lf ", A[i*incRowA+j*incColA]);
            } else {
                printf("%10.4lf ", 0.0);
            }
        }
        printf("\n");
    }
    printf("\n\n");
}

//-- some BLAS Level 1 procedures and functions --------------------------------

double
ddot(int n, const double *x, int incX, const double *y, int incY)
{
    int     i;
    double  alpha = 0;

    for (i=0; i<n; ++i) {
        alpha += x[i*incX]*y[i*incY];

    }
    return alpha;
}

void
daxpy(int n, double alpha, const double *x, int incX, double *y, int incY)
{
    int i;

    if (alpha==0) {
        return;
    }
    for (i=0; i<n; ++i) {
        y[i*incY] += alpha*x[i*incX];
    }
}

void
dscal(int n, double alpha, double *x, int incX)
{
    int i;

    if (alpha==1) {
        return;
    }
    for (i=0; i<n; ++i) {
        x[i*incX] *= alpha;
    }
}

void
dcopy(int n, const double *x, int incX, double *y, int incY)
{
    int i;

    for (i=0; i<n; ++i) {
        y[i*incY] = x[i*incX];
    }
}

void
dswap(int n, double *x, int incX, double *y, int incY)
{
    int i;

    for (i=0; i<n; ++i) {
        double tmp;

        tmp = x[i*incX];
        x[i*incX] = y[i*incY];
        y[i*incY] = tmp;
    }
}

double
damax(int n, const double *x, int incX)
{
    int     i;
    double  result = 0;

    for (i=0; i<n; ++i) {
        if (fabs(x[i*incX])>result) {
            result = fabs(x[i*incX]);
        }
    }
    return result;
}

double
dgenrm1(int m, int n, const double *A, int incRowA, int incColA)
{
    int     i, j;
    double  result = 0;

    for (j=0; j<n; ++j) {
        double sum = 0;
        for (i=0; i<m; ++i) {
            sum += fabs(A[i*incRowA+j*incColA]);
        }
        if (sum>result) {
            result = sum;
        }
    }
    return result;
}

double
dtrnrm1(int m, int n,  int unitDiag, int lower,
        const double *A, int incRowA, int incColA)
{
    int     i, j;
    double  result = 0;

    for (j=0; j<n; ++j) {
        double sum = 0;
        for (i=0; i<m; ++i) {
            if (unitDiag && (i==j)) {
                sum += 1.0;
            } else if ((lower && (i>=j)) || (!lower && (i<=j))) {
                sum += fabs(A[i*incRowA+j*incColA]);
            }
        }
        if (sum>result) {
            result = sum;
        }
    }
    return result;
}

//-- error bounds --------------------------------------------------------------

double
err_dgemv(int m, int n, double alpha,
          const double *A, int incRowA, int incColA,
          const double *x, int incX,
          double beta,
          const double *y0, int incY0,
          double *y1, int incY1)
{
    int    max_mn = (m>n) ? m : n;
    double normA  = fabs(alpha)*dgenrm1(m, n, A, incRowA, incColA);
    double normX  = damax(n, x, incX);
    double normY0 = fabs(beta)*damax(m, y0, incY0);
    double normD;

    daxpy(m, -1.0, y0, incY0, y1, incY1);
    normD = damax(n, y1, incY1);

    return normD/(normA*normX*normY0*max_mn);
}

double
err_dtrmv(int n, int unitDiag, int lower,
          const double *A, int incRowA, int incColA,
          const double *x0, int incX0,
          double *x1, int incX1)
{
    double normA  = dtrnrm1(n, n, unitDiag, lower, A, incRowA, incColA);
    double normX0 = damax(n, x0, incX0);
    double normD;

    daxpy(n, -1.0, x0, incX0, x1, incX1);
    normD = damax(n, x1, incX1);

    return normD/(n*normA*normX0);
}

double
err_dtrsv(int n, int unitDiag, int lower,
          const double *A, int incRowA, int incColA,
          const double *x0, int incX0,
          double *x1, int incX1)
{
    double normA  = dtrnrm1(n, n, unitDiag, lower, A, incRowA, incColA);
    double normX0 = damax(n, x0, incX0);
    double normD;

    daxpy(n, -1.0, x0, incX0, x1, incX1);
    normD = damax(n, x1, incX1);

    return normD/(n*normA*normX0);
}

//-- Fused BLAS Level 1 procedures and functions -------------------------------

#ifndef DGEMV_MxBS
#define DGEMV_MxBS 4
#endif

void
dgemv_mxBS(int m, double alpha,
           const double *A, int incRowA, int incColA,
           const double *x, int incX,
           double *y, int incY)
{
    int i, j;

    if (alpha==0) {
        return;
    }
    for (i=0; i<m; ++i) {
        double dot = 0;
        for (j=0; j<DGEMV_MxBS; ++j) {
            dot += A[i*incRowA+j*incColA]*x[j*incX];
        }
        y[i*incY] += alpha*dot;
    }
}

#ifndef DGEMV_BSxN
#define DGEMV_BSxN 4
#endif

void
dgemv_BSxn(int n, double alpha,
           const double *A, int incRowA, int incColA,
           const double *x, int incX,
           double beta,
           double *y, int incY)
{
    int i, j;

    if (alpha==0) {
        return;
    }
    if (beta!=1) {
        for (i=0; i<DGEMV_BSxN; ++i) {
            y[i*incY] *= beta;
        }
    }
    for (j=0; j<n; ++j) {
        for (i=0; i<DGEMV_BSxN; ++i) {
            y[i*incY] += alpha*A[i*incRowA+j*incColA]*x[j*incX];
        }
    }
}

//-- GEMV implementations ------------------------------------------------------

void
dgemv_ref(int m, int n, double alpha,
          const double *A, int incRowA, int incColA,
          const double *x, int incX,
          double beta,
          double *y, int incY)
{
    int i, j;

    for (i=0; i<m; ++i) {
        if (beta==0) {
            y[i*incY] = 0;
        } else {
            y[i*incY] *= beta;
        }
        for (j=0; j<n; ++j) {
            y[i*incY] += alpha*A[i*incRowA+j*incColA]*x[j*incX];
        }
    }
}

void
dgemv_unblk(int m, int n, double alpha,
            const double *A, int incRowA, int incColA,
            const double *x, int incX,
            double beta,
            double *y, int incY)
{
    if (incRowA<incColA) {
        int j;

        dscal(m, beta, y, incY);

        for (j=0; j<n; ++j) {
            daxpy(m, alpha*x[j*incX], &A[j*incColA], incRowA, y, incY);
        }
    } else {
        int i;

        for (i=0; i<m; ++i) {
            y[i*incY] *= beta;
            y[i*incY] += alpha*ddot(n, &A[i*incRowA], incColA, x, incX);
        }
    }
}

void
dgemv_ulm(int m, int n, double alpha,
          const double *A, int incRowA, int incColA,
          const double *x, int incX,
          double beta,
          double *y, int incY)
{
    if (incRowA<incColA) {
        int bs = DGEMV_MxBS;
        int j;

        dscal(m, beta, y, incY);

        for (j=0; j+bs<=n; j+=bs) {
            dgemv_mxBS(m, alpha,
                       &A[j*incColA], incRowA, incColA,
                       &x[j*incX], incX,
                       y, incY);
        }
        dgemv_unblk(m, n-j, alpha,
                    &A[j*incColA], incRowA, incColA,
                    &x[j*incX], incX,
                    1.0,
                    y, incY);
    } else {
        int bs = DGEMV_BSxN;
        int i;

        for (i=0; i+bs<=m; i+=bs) {
            dgemv_BSxn(n, alpha,
                       &A[i*incRowA], incRowA, incColA,
                       x, incX,
                       beta,
                       &y[i*incY], incY);
        }
        dgemv_unblk(m-i, n, alpha,
                    &A[i*incRowA], incRowA, incColA,
                    x, incX,
                    beta,
                    &y[i*incY], incY);
    }
}

//-- TRMV implementations ------------------------------------------------------

void
dtrlmv_ref(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    int i, j;

    for (j=n-1; j>=0; --j) {
        for (i=j+1; i<n; ++i) {
            x[i*incX] += A[i*incRowA+j*incColA]*x[j*incX];
        }
        if (!unitDiag) {
            x[j*incX] *= A[j*(incRowA+incColA)];
        }
    }
}

void
dtrlmv_col(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    int k;

    if (unitDiag) {
        for (k=n-1; k>=0; --k) {
            daxpy(n-k-1, x[k*incX],
                  &A[(k+1)*incRowA+k*incColA], incRowA,
                  &x[(k+1)*incX], incX);
        }
    } else {
        for (k=n-1; k>=0; --k) {
            daxpy(n-k-1, x[k*incX],
                  &A[(k+1)*incRowA+k*incColA], incRowA,
                  &x[(k+1)*incX], incX);
            x[k*incX] *= A[k*(incRowA+incColA)];
        }
    }
}

void
dtrlmv_row(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    int k;

    if (unitDiag) {
        for (k=n-1; k>=0; --k) {
            x[k*incX] += ddot(k, &A[k*incRowA], incColA, x, incX);
        }
    } else {
        for (k=n-1; k>=0; --k) {
            x[k*incX] = ddot(k+1, &A[k*incRowA], incColA, x, incX);
        }
    }
}

void
dtrlmv_ulm(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    if (incRowA<incColA) {
        int bs = DGEMV_MxBS;
        int k  = (n/bs)*bs;

        dtrlmv_col(n-k, unitDiag,
                   &A[k*(incRowA+incColA)], incRowA, incColA,
                   &x[k*incX], incX);

        for (k-=bs; k>=0; k-=bs) {
            dgemv_mxBS(n-k-bs, 1.0,
                       &A[(k+bs)*incRowA+k*incColA], incRowA, incColA,
                       &x[k*incX], incX,
                       &x[(k+bs)*incX], incX);
            dtrlmv_col(bs, unitDiag,
                       &A[k*(incRowA+incColA)], incRowA, incColA,
                       &x[k*incX], incX);
        }
    } else {
        double buffer[DGEMV_BSxN];
        int    bs = DGEMV_BSxN;
        int    k  = (n/bs)*bs;

        dgemv_ulm(n-k, k, 1.0,
                  &A[k*incRowA], incRowA, incColA,
                  x, incX,
                  0.0,
                  buffer,1);
        dtrlmv_row(n-k, unitDiag,
                   &A[k*(incRowA+incColA)], incRowA, incColA,
                   &x[k*incX], incX);
        daxpy(n-k, 1.0, buffer, 1, &x[k*incX], incX);

        for (k-=bs; k>=0; k-=bs) {
            dgemv_BSxn(k, 1.0,
                       &A[k*incRowA], incRowA, incColA,
                       x, incX,
                       0.0,
                       buffer, 1);
            dtrlmv_col(bs, unitDiag,
                       &A[k*(incRowA+incColA)], incRowA, incColA,
                       &x[k*incX], incX);
            daxpy(bs, 1.0, buffer, 1, &x[k*incX], incX);
        }
    }
}

//-- TRSV implementations ------------------------------------------------------

void
dtrlsv_ref(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    int i, j;

    for (i=0; i<n; ++i) {
        for (j=0; j<i; ++j) {
            x[i*incX] -= A[i*incRowA+j*incColA]*x[j*incX];
        }
        x[i*incX] /= (unitDiag) ? 1.0 : A[i*(incRowA+incColA)];
    }
}

void
dtrlsv_row(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    // ... your code here ....
}

void
dtrlsv_col(int n, int unitDiag,
           const double *A, int incRowA, int incColA,
           double *x, int incX)
{
    // ... your code here ....
}

//-- Wrapper for the Intel MKL -------------------------------------------------

#ifdef USE_MKL
#include <mkl_types.h>

// pre declaration so we don't need to include the complete mkl_blas header.
void
dgemv(const char *trans,
      const MKL_INT *m, const MKL_INT *n, const double *alpha,
      const double *A, const MKL_INT *ldA, const double *x,
      const MKL_INT *incX,
      const double *beta, double *y, const MKL_INT *incY);

void
dgemv_mkl(MKL_INT m, MKL_INT n,
          double alpha,
          const double *A, MKL_INT incRowA, MKL_INT incColA,
          const double *x, MKL_INT incX,
          double beta,
          double *y, MKL_INT incY)
{
    MKL_INT ldA   = (incRowA==1) ? incColA : incRowA;
    char    trans = (incRowA==1) ? 'N' : 'T';
    MKL_INT M     = (incRowA==1) ? m : n;
    MKL_INT N     = (incRowA==1) ? n : m;

    dgemv(&trans, &M, &N, &alpha, A, &ldA, x, &incX, &beta, y, &incY);
}

void dtrmv(const char *uplo, const char *transa, const char *diag,
           const MKL_INT *n,
           const double *a, const MKL_INT *lda,
           double *b, const MKL_INT *incx);

void
dtrmv_mkl(int lower, int unitDiag, MKL_INT n,
          const double *A, int incRowA, int incColA,
          double *x, int incX)
{
    MKL_INT ldA   = (incRowA==1) ? incColA : incRowA;

    dtrmv(&N, A, &ldA, x, &incX);
}

#endif // USE_MKL

//------------------------------------------------------------------------------

#ifndef MIN_N
#define MIN_N 100
#endif

#ifndef MAX_N
#define MAX_N 8000
#endif

#ifndef INC_N
#define INC_N 100
#endif

#ifndef ROWMAJOR
#define ROWMAJOR 0
#endif

#ifndef UNITDIAG
#define UNITDIAG 0
#endif

#if (ROWMAJOR==1)
#   define INCROW_A  MAX_N
#   define INCCOL_A  1
#else
#   define INCROW_A  1
#   define INCCOL_A  MAX_N
#endif

#ifndef INC_X
#define INC_X 1
#endif

double A[MAX_N*MAX_N];
double x_[MAX_N];
double x_0[INC_X*MAX_N];
double x_1[INC_X*MAX_N];

int
main()
{
    int n;

    randGeMatrix(MAX_N, MAX_N, A, INCROW_A, INCCOL_A);
    makeTrlDiagDom(MAX_N, UNITDIAG, A, INCROW_A, INCCOL_A);
    randGeMatrix(MAX_N, 1, x_, 1, 0);

    printf("#UNITDIAG=%3d\n", UNITDIAG);
    printf("# %51s %25s %38s\n", "dtrlsv_ref", "dtrlsv_row", "dtrlsv_col");
    printf("#%9s %9s %9s", "n", "INCROW_A", "INCCOL_A");
    printf(" %9s", "INC_X");
    printf(" %12s %12s", "t", "MFLOPS");
    printf(" %12s %12s %12s", "t", "MFLOPS", "err");
    printf(" %12s %12s %12s", "t", "MFLOPS", "err");
    printf("\n");

    for (n=MIN_N; n<=MAX_N; n+=INC_N) {
        int     runs = 1;
        int     ops = n*(n+1)/2 + (n-1)*n/2;
        double  t0, t1, dt, err;

        printf(" %9d %9d %9d", n, INCROW_A, INCCOL_A);
        printf(" %9d", INC_X);

        t0   = 0;
        runs = 0;
        do {
            dcopy(n, x_, 1, x_0, INC_X);
            dt = walltime();
            dtrlsv_ref(n, UNITDIAG,
                       A, INCROW_A, INCCOL_A,
                       x_0, INC_X);
            dt = walltime() - dt;
            t0 += dt;
            ++runs;
        } while (t0<0.3);
        t0 /= runs;

        printf(" %12.2e %12.2lf", t0, ops/(1000*1000*t0));

        t1   = 0;
        runs = 0;
        do {
            dcopy(n, x_, 1, x_1, INC_X);
            dt = walltime();
            dtrlsv_row(n, UNITDIAG,
                       A, INCROW_A, INCCOL_A,
                       x_1, INC_X);
            dt = walltime() - dt;
            t1 += dt;
            ++runs;
        } while (t1<0.3);
        t1 /= runs;

        err = err_dtrsv(n, UNITDIAG, 1,
                        A, INCROW_A, INCCOL_A,
                        x_0, INC_X,
                        x_1, INC_X);

        printf(" %12.2e %12.2lf %12.2e", t1, ops/(1000*1000*t1), err);

        t1   = 0;
        runs = 0;
        do {
            dcopy(n, x_, 1, x_1, INC_X);
            dt = walltime();
            dtrlsv_col(n, UNITDIAG,
                       A, INCROW_A, INCCOL_A,
                       x_1, INC_X);
            dt = walltime() - dt;
            t1 += dt;
            ++runs;
        } while (t1<0.3);
        t1 /= runs;

        err = err_dtrsv(n, UNITDIAG, 1,
                        A, INCROW_A, INCCOL_A,
                        x_0, INC_X,
                        x_1, INC_X);

        printf(" %12.2e %12.2lf %12.2e", t1, ops/(1000*1000*t1), err);


        printf("\n");
    }

    return 0;
}