linalg3.h

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#ifndef VA_LINALG3_H
#define VA_LINALG3_H
 
#include "common_defs.h"
 
/***************************************************************************/
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_sign(VA_REAL a, VA_REAL b)
{
    return (b < (VA_REAL) 0.) ? -fabs(a) : fabs(a);
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_cabs(VA_DOUBLE_COMPLEX a)
{
#if defined(CUDA)
    return abs(a);
#elif defined(HIP)
    VA_REAL ar = hipCreal(a);
    VA_REAL ai = hipCimag(a);
    return sqrt(ar*ar + ai*ai);
#else
    return cabs(a);
#endif
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_DOUBLE_COMPLEX valib_cmultiply(VA_DOUBLE_COMPLEX a, VA_DOUBLE_COMPLEX b)
{
#if defined(CUDA)
    return a*b;
#elif defined(HIP)
    double ar = hipCreal(a);
    double ai = hipCimag(a);
    double br = hipCreal(b);
    double bi = hipCimag(b);
 
    double cr = ar*br - ai*bi;
    double ci = ar*bi + ai*br;
    VA_DOUBLE_COMPLEX c = {cr,ci};
    return c;
#else
    return a*b;
#endif
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_cubic_root(VA_REAL x)
{
    return valib_sign(pow(fabs(x), 1.0/3.0), x);
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_max(VA_REAL a, VA_REAL b)
{
    return (a > b) ? a : b;
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_min(VA_REAL a, VA_REAL b)
{
    return (a < b) ? a : b;
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_square(VA_REAL a)
{
    return (a * a);
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_swap_values(VA_REAL *a, VA_REAL *b)
{
    VA_REAL buf;
    buf = *a;
    *a  = *b;
    *b  = buf;
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_solve_cubic_equation(VA_REAL a, VA_REAL b,
                                              VA_REAL c, VA_REAL d,
                                              VA_COMPLEX *x1,
                                              VA_COMPLEX *x2,
                                              VA_COMPLEX *x3,
                                              int *info)
{
    int debug = 0;
 
    // This computation only works in double precision.
    double ad = a;
    double bd = b;
    double cd = c;
    double dd = d;
 
    double q = pow(bd/(3.*ad),3) - (bd*cd)/(6.*ad*ad) + dd/(2.*ad);
    double p = (3*ad*cd - bd*bd) / (9*ad*ad);
 
    double delta = - (p*p*p + q*q);
 
    if (debug) {
        printf("q:     %lf  \n", q);
        printf("p:     %lf  \n", p);
        printf("delta: %lf  \n", delta);
    }
 
    if (delta <= 0) {
        if (debug) {
           printf("This is a good case with two complex roots.\n");
        }
    }
    else {
        if (debug) {
           printf("Discriminant is positive, this is not a good way to get roots.\n");
        }
        *info = -1;
        return;
    }
 
    double sr = sqrt(-delta);
 
    if (debug) {
        printf("sr:     %lf  \n", sr);
    }
 
    double mqpsr = -q + sr;
    double mqmsr = -q - sr;
 
    if (debug) {
        printf("mqpsr:     %lf  \n", mqpsr);
        printf("mqmsr:     %lf  \n", mqmsr);
    }
 
    double u = valib_cubic_root(mqpsr);
    double v = valib_cubic_root(mqmsr);
 
    if (debug) {
        printf("fabs(u*v + p):     %lf  \n", fabs(u*v + p));
    }
    if (fabs(u*v + p) > 1.e-4) {
        if (debug) {
            printf("Check of uv = -p failed, check = %lf.", fabs(u*v + p));
        }
        *info = -2;
        return;
    }
 
// define imaginary unit for CUDA or HIP
#if defined(CUDA)
    VA_DOUBLE_COMPLEX e1(-0.5,  sqrt(3.)/2.);
    VA_DOUBLE_COMPLEX e2(-0.5, -sqrt(3.)/2.);
 
    VA_DOUBLE_COMPLEX uc = u;
    VA_DOUBLE_COMPLEX vc = v;
#elif defined(HIP)
    VA_DOUBLE_COMPLEX e1 = {-0.5,  sqrt(3.)/2.};
    VA_DOUBLE_COMPLEX e2 = {-0.5, -sqrt(3.)/2.};
 
    VA_DOUBLE_COMPLEX uc = {u, 0};
    VA_DOUBLE_COMPLEX vc = {v, 0};
#else
    VA_DOUBLE_COMPLEX e1 = -0.5 + sqrt(3.)/2.*I;
    VA_DOUBLE_COMPLEX e2 = -0.5 - sqrt(3.)/2.*I;
 
    VA_DOUBLE_COMPLEX uc = u + 0.*I;
    VA_DOUBLE_COMPLEX vc = v + 0.*I;
#endif
 
    VA_DOUBLE_COMPLEX y1 = uc + vc;
    VA_DOUBLE_COMPLEX y2 = valib_cmultiply(e1,uc) + valib_cmultiply(e2,vc);
    VA_DOUBLE_COMPLEX y3 = valib_cmultiply(e2,uc) + valib_cmultiply(e1,vc);
 
    double shift = -bd/(3.*ad);
#if defined(CUDA)
    VA_DOUBLE_COMPLEX shiftc = shift;
#elif defined(HIP)
    VA_DOUBLE_COMPLEX shiftc = {shift,0};
#else
    VA_DOUBLE_COMPLEX shiftc = shift + 0.*I;
#endif
 
    *x1 = y1 + shiftc;
    *x2 = y2 + shiftc;
    *x3 = y3 + shiftc;
 
#if defined(CUDA)
    VA_DOUBLE_COMPLEX ac = a;
    VA_DOUBLE_COMPLEX bc = b;
    VA_DOUBLE_COMPLEX cc = c;
    VA_DOUBLE_COMPLEX dc = d;
#elif defined(HIP)
    VA_DOUBLE_COMPLEX ac = {a,0};
    VA_DOUBLE_COMPLEX bc = {b,0};
    VA_DOUBLE_COMPLEX cc = {c,0};
    VA_DOUBLE_COMPLEX dc = {d,0};
#else
    VA_DOUBLE_COMPLEX ac = a + 0.*I;
    VA_DOUBLE_COMPLEX bc = b + 0.*I;
    VA_DOUBLE_COMPLEX cc = c + 0.*I;
    VA_DOUBLE_COMPLEX dc = d + 0.*I;
#endif
 
    // check that ax^3 + bx^2 + cx + d is close to zero for the roots
    VA_DOUBLE_COMPLEX check1 = valib_cmultiply(ac,valib_cmultiply(valib_cmultiply((*x1),(*x1)),(*x1))) + valib_cmultiply(bc,valib_cmultiply((*x1),(*x1))) + valib_cmultiply(cc,(*x1)) + dc;
    VA_DOUBLE_COMPLEX check2 = valib_cmultiply(ac,valib_cmultiply(valib_cmultiply((*x2),(*x2)),(*x2))) + valib_cmultiply(bc,valib_cmultiply((*x2),(*x2))) + valib_cmultiply(cc,(*x2)) + dc;
    VA_DOUBLE_COMPLEX check3 = valib_cmultiply(ac,valib_cmultiply(valib_cmultiply((*x3),(*x3)),(*x3))) + valib_cmultiply(bc,valib_cmultiply((*x3),(*x3))) + valib_cmultiply(cc,(*x3)) + dc;
 
    if (valib_cabs(check1) > 1.e-5) {
        if (debug) {
            printf("Check of x1 failed.");
        }
    }
    if (valib_cabs(check2) > 1.e-5) {
        if (debug) {
            printf("Check of x2 failed.");
        }
    }
    if (valib_cabs(check3) > 1.e-5) {
        if (debug) {
            printf("Check of x3 failed.");
        }
    }
 
    // finished correctly
    *info = 0;
}
 
/***************************************************************************/
/***************************************************************************/
VA_DEVICE_FUN void valib_vec_zero3(VA_REAL *v)
{
    int i;
    for (i = 0; i < 3; i++) {
        v[i] = 0.;
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN VA_REAL valib_scalar_prod3(VA_REAL *v1, VA_REAL *v2)
{
    VA_REAL result = (VA_REAL) 0.;
    int i;
    for (i = 0; i < 3; i++)
    {
        result = result + v1[i]*v2[i];
    }
    return result;
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_cross_prod3(VA_REAL *v1, VA_REAL *v2, VA_REAL *v3)
{
    v3[0] = v1[1] * v2[2] - v1[2] * v2[1];
    v3[1] = v1[2] * v2[0] - v1[0] * v2[2];
    v3[2] = v1[0] * v2[1] - v1[1] * v2[0];
}
 
/***************************************************************************/
/***************************************************************************/
VA_DEVICE_FUN void valib_mat_zero3(VA_REAL *A)
{
    int i, j;
    for (j = 0; j < 3; j++) {
        for (i = 0; i < 3; i++) {
            A[j*3+i] = 0.;
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_mat_copy3(VA_REAL *A, VA_REAL *B)
{
    int i, j;
    for (j = 0; j < 3; j++) {
        for (i = 0; i < 3; i++) {
            B[j*3+i] = A[j*3+i];
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_norm_frobenius3(VA_REAL *A, VA_REAL *norm)
{
    *norm = 0.0;
    int i, j;
    for (j = 0; j < 3; j++) {
        for (i = 0; i < 3; i++) {
            *norm = *norm + A[j*3+i]*A[j*3+i];
        }
    }
    *norm = sqrt(*norm);
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_trace3(VA_REAL *A, VA_REAL *tr3)
{
    *tr3 = A[0] + A[4] + A[8];
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_second_invariant3(VA_REAL *A,
                                           VA_REAL *second_invariant)
{
    *second_invariant = ( A[0]*A[4] + A[4]*A[8] + A[0]*A[8] )
                      - ( A[1]*A[3] + A[2]*A[6] + A[5]*A[7] );
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_determinant3(VA_REAL *A, VA_REAL *det)
{
    *det = A[0] * A[4] * A[8]
         + A[3] * A[7] * A[2]
         + A[6] * A[1] * A[5]
         - A[0] * A[7] * A[5]
         - A[3] * A[1] * A[8]
         - A[6] * A[4] * A[2];
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_transpose3(VA_REAL *A)
{
    valib_swap_values(&A[1], &A[3]);
    valib_swap_values(&A[2], &A[6]);
    valib_swap_values(&A[5], &A[7]);
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_sym_antisym3(VA_REAL *A)
{
    VA_REAL ud, ld;
 
    int i, j;
    for (i = 0; i < 3; i++) {
        for (j = i+1; j < 3; j++) {
            ud = A[i + j*3];
            ld = A[j + i*3];
 
            A[i + j*3] = (VA_REAL) 0.5 * (ud + ld);
            A[j + i*3] = (VA_REAL) 0.5 * (ld - ud);
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_deviatorise3(VA_REAL *A)
{
    // get trace(A)
    VA_REAL trace;
    valib_trace3(A, &trace);
 
    // subtract a multiple of the third of the trace from the diagonal
    A[0] = A[0] - (VA_REAL) (1./3.) * trace;
    A[4] = A[4] - (VA_REAL) (1./3.) * trace;
    A[8] = A[8] - (VA_REAL) (1./3.) * trace;
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_matvec3_prod(int trans_a, VA_REAL *A,
                                      VA_REAL *x, VA_REAL *y)
{
    int i, j;
    for (i = 0; i < 3; i++) {
        y[i] = 0.0;
        for (j = 0; j < 3; j++) {
            if (trans_a == 0) {
                y[i] = y[i] + A[j*3+i]*x[j];
            }
            else {
                y[i] = y[i] + A[i*3+j]*x[j];
            }
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_rank1_update3(VA_REAL alpha,
                                       VA_REAL *x, VA_REAL *y,
                                       VA_REAL *A)
{
    int ind;
    VA_REAL tmp;
 
    ind = 0;
    int i, j;
    for (j = 0; j < 3; j++) {
        tmp = alpha * y[j];
        for (i = 0; i < 3; i++) {
            A[ind] = A[ind] + x[i]*tmp;
            ind = ind + 1;
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_tatt3(VA_REAL *A, VA_REAL *T, VA_REAL *TATT)
{
    unsigned i, j, ind;
 
    // zero TATT
    valib_mat_zero3(TATT);
 
    // perform series of rank one updates
    ind = 0;
    for (i = 0; i < 3; i++) {
        for (j = 0; j < 3; j++) {
            // call rank one update on each element of A
            valib_rank1_update3(A[ind], &T[j*3], &T[i*3], TATT);
            ind = ind + 1;
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_matmat3_prod(int trans_a, int trans_b,
                                      VA_REAL *A, VA_REAL *B, VA_REAL *C)
{
    int i, j, k;
    for (i = 0; i < 3; i++) {
        for (j = 0; j < 3; j++) {
            for (k = 0; k < 3; k++) {
                if      (trans_a == 0 && trans_b == 0) {
                    C[j*3+i] = C[j*3+i] + A[k*3+i] * B[j*3+k];
                }
                else if (trans_a != 0 && trans_b == 0) {
                    C[j*3+i] = C[j*3+i] + A[i*3+k] * B[j*3+k];
                }
                else if (trans_a == 0 && trans_b != 0) {
                    C[j*3+i] = C[j*3+i] + A[k*3+i] * B[k*3+j];
                }
                else {//(trans_a != 0 && trans_b != 0)
                    C[j*3+i] = C[j*3+i] + A[i*3+k] * B[k*3+j];
                }
            }
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_ttat3(VA_REAL *A, VA_REAL *T, VA_REAL *TTAT)
{
    // zero TATT
    valib_mat_zero3(TTAT);
 
    // first multiply A*T
    VA_REAL AT[3*3];
    valib_mat_zero3(AT);
    valib_matmat3_prod(0, 0, A, T, AT);
 
    // now premultiply T^T * (A*T)
    valib_matmat3_prod(1, 0, T, AT, TTAT);
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_gram_schmidt3(VA_REAL *Q)
{
    VA_REAL factor;
 
    // column-wise process
    int i, j, k;
    for (j = 0; j < 3; j++) {
        for (i = 0; i < j; i++) {
            // compute the scalar product with previous columns
            factor = valib_scalar_prod3(&Q[j*3], &Q[i*3]);
 
            // subtract the projected vector
            for (k = 0; k < 3; k++) {
                Q[j*3+k] = Q[j*3+k] - factor * Q[i*3+k];
            }
        }
 
        // normalize the column
        factor = valib_scalar_prod3(&Q[j*3], &Q[j*3]);
        for (k = 0; k < 3; k++) {
            Q[j*3+k] = Q[j*3+k] / sqrt(factor);
        }
    }
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_eigenvalues_sym3(VA_REAL *A, VA_REAL *eigval)
{
    // compute first invariant
    VA_REAL Idev;
    valib_trace3(A, &Idev);
    VA_REAL IdevMod = Idev / 3.;
 
    // compute deviator
    VA_REAL dev[9];
    valib_mat_copy3(A, dev);
    dev[0] = dev[0] - IdevMod;
    dev[4] = dev[4] - IdevMod;
    dev[8] = dev[8] - IdevMod;
 
    // compute second invariant of deviator
    VA_REAL IIdev;
    valib_second_invariant3(dev, &IIdev);
    VA_REAL IIdevMod = -IIdev / 3.;
 
    // compute half the determinant (third invariant) of deviator
    VA_REAL IIIdev;
    valib_determinant3(dev, &IIIdev);
    VA_REAL IIIdevMod = 0.5 * IIIdev;
 
    // compute the phi angle
    VA_REAL aux = IIdevMod*IIdevMod*IIdevMod - IIIdevMod*IIIdevMod;
    // assure nonnegativity
    if (aux < 0.) {
        aux = 0.;
    }
    VA_REAL phi;
    phi = atan2( sqrt(aux), IIIdevMod ) / 3.;
    // readjust to the right quadrant
    if (phi < 0.) phi += VA_PI / 3.;
 
    // eigenvalues
    eigval[0] = IdevMod + 2.*sqrt(IIdevMod) * cos(phi);
    eigval[1] = IdevMod - sqrt(IIdevMod) * (cos(phi) + sqrt(3.) * sin(phi));
    eigval[2] = IdevMod - sqrt(IIdevMod) * (cos(phi) - sqrt(3.) * sin(phi));
 
    // sort eigenvalues
    if (eigval[0] > eigval[1] ) valib_swap_values( &eigval[0],  &eigval[1] );
    if (eigval[1] > eigval[2] ) valib_swap_values( &eigval[1],  &eigval[2] );
    if (eigval[0] > eigval[1] ) valib_swap_values( &eigval[0],  &eigval[1] );
 
    return;
}
 
/***************************************************************************/
VA_DEVICE_FUN int valib_find_pivot3(VA_REAL *A, int jcol, int start)
{
    int debug = 0;
    int ipiv = start;
    for (int i = start+1; i<3; i++) {
        if (fabs(A[i+jcol*3]) > fabs(A[ipiv+jcol*3]))
            ipiv = i;
    }
 
    VA_REAL numerical_zero = 1.0e-5;
    if (fabs(A[ipiv+jcol*3]) < numerical_zero) {
        if (debug)
            printf("Warning: no nonzero pivot in column %d.\n", jcol);
 
        ipiv = -1;
    }
 
    return ipiv;
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_find_nullspace3(VA_REAL *A, VA_REAL *x, int *info)
{
    int debug = 0;
 
    // zero the vector
    x[0] = 0.; x[1] = 0.; x[2] = 0.;
 
    // handle special cases with zero column
    int num_zero_cols = 0;
    int i_zero_col = -1;
    int ip;
    for (int ji = 0; ji < 3; ji++) {
        int j = 2-ji;
        ip = valib_find_pivot3(A,j,0);
        if (ip == -1) {
            if (debug) {
               printf("There is no meaningful pivot in the %d-th column.\n",j);
            }
            num_zero_cols++;
            i_zero_col = j;
        }
    }
 
    if (num_zero_cols > 0) {
        if (num_zero_cols > 1) {
            if (debug) {
               printf("Oops! The matrix has more than one zero columns. I search only for 1-dimensional nullspace.\n");
            }
            *info = -1;
            return;
        }
        else {
            x[i_zero_col] = 1.;
            *info = 0;
            return;
        }
    }
 
    // in the usual case, permute the rows and eliminate first column
    if (ip != 0) {
        for (int j = 0; j < 3; j++) {
            valib_swap_values(&A[j*3], &A[ip+j*3]);
        }
    }
 
    if (debug) {
        printf("Matrix in GEM after 1st swap: \n");
        for (int i = 0; i < 3; i++) {
            for (int j = 0; j < 3; j++) {
                printf(" %lf ", A[i+j*3]);
            }
            printf("\n");
        }
    }
 
    VA_REAL coef;
 
    // eliminate the first column using the pivot
    for (int i = 1; i < 3; i++) {
        coef = A[i] / A[0];
        for (int j = 0; j < 3; j++) {
            A[i+j*3] -= coef*A[j*3];
        }
    }
 
    if (debug) {
        printf("Matrix in GEM without 1st col: \n");
        for (int i = 0; i < 3; i++) {
            for (int j = 0; j < 3; j++) {
                printf(" %lf ", A[i+j*3]);
            }
            printf("\n");
        }
    }
 
    // now work on the second column
    int jpiv, jfree;
    ip = valib_find_pivot3(A,1,1);
    if (ip == -1) {
        // the second column is not a pivot one, we can search for the nullspace
        jpiv  = 2;
        jfree = 1;
    }
    else {
        jpiv  = 1;
        jfree = 2;
    }
 
    ip = valib_find_pivot3(A,jpiv,1);
    if (ip == -1) {
        if (debug) {
           printf("Something is wrong with the matrix, the defect seems larger than one.");
        }
        *info = -2;
        return;
    }
    if (ip == 2) {
        // swap the rows
        for (int j = 0; j < 3; j++) {
            valib_swap_values(&A[1+j*3], &A[2+j*3]);
        }
    }
 
    // eliminate the last row
    coef = A[2+jpiv*3] / A[1+jpiv*3];
    for (int j = 0; j < 3; j++) {
        A[2+j*3] -= coef*A[1+j*3];
    }
    // eliminate the first row
    coef = A[0+jpiv*3] / A[1+jpiv*3];
    for (int j = 0; j < 3; j++) {
        A[j*3] -= coef*A[1+j*3];
    }
 
    // check that the last row, in particular the last entry, is zero
    if (valib_find_pivot3(A,2,2) != -1) {
        if (debug) {
            printf("Something is wrong, the matrix seems to be full rank.\n");
        }
        *info = -2;
        return;
    }
 
    if (debug) {
        printf("Matrix in GEM before dividing by diagonal: \n");
        for (int i = 0; i < 3; i++) {
            for (int j = 0; j < 3; j++) {
                printf(" %lf ", A[i+j*3]);
            }
            printf("\n");
        }
    }
 
    // finalize the Gauss-Jordan elimination by dividing with the diagonal values
    coef = A[0];
    for (int j = 0; j < 3; j++) {
        A[j*3]   /= coef;
    }
    coef = A[1+jpiv*3];
    for (int j = 0; j < 3; j++) {
        A[1+j*3] /= coef;
    }
 
    if (debug) {
        printf("Matrix in GEM in the upper right echelon form:\n");
        for (int i = 0; i < 3; i++) {
            for (int j = 0; j < 3; j++) {
                printf(" %d %lf ", i+j*3, A[i+j*3]);
            }
            printf("\n");
        }
    }
 
    // fill the nullspace basis vector
    x[jfree] = 1.;
    x[jpiv]  = -A[jpiv+jfree*3];
    x[0]     = -A[jfree*3];
 
    // normalize the basis vector
    VA_REAL norm = sqrt(valib_scalar_prod3(&x[0], &x[0]));
    for (int i = 0; i < 3; i++) {
        x[i] /= norm;
    }
 
    *info = 0;
}
 
 
/***************************************************************************/
VA_DEVICE_FUN void valib_constructQ3(VA_REAL sina, VA_REAL cosa,
                                     VA_REAL sinb, VA_REAL cosb,
                                     VA_REAL sing, VA_REAL cosg,
                                     VA_REAL *Q )
{
    Q[0] = cosa*cosb*cosg - sina*sing;
    Q[1] = -cosa*cosb*sing - sina*cosg;
    Q[2] = cosa*sinb;
    Q[3] = sina*cosb*cosg + cosa*sing;
    Q[4] = -sina*cosb*sing + cosa*cosg;
    Q[5] = sina*sinb;
    Q[6] = -sinb*cosg;
    Q[7] = sinb*sing;
    Q[8] = cosb;
}
 
/***************************************************************************/
VA_DEVICE_FUN void valib_constructQfromN3(VA_REAL *n, VA_REAL *Q)
{
    VA_REAL vec[3];
    VA_REAL factor;
 
    // initialize the first row of Q with n
    Q[0] = n[0]; Q[1] = n[1]; Q[2] = n[2];
 
    // fill some entries to initialize the other entries in matrix Q
    Q[3] = n[1]; Q[4] = n[2]; Q[5] = n[0];
    Q[6] = n[2]; Q[7] = n[0]; Q[8] = n[1];
 
    // perform Gram-Schmidt orthogonalization
    valib_gram_schmidt3(Q);
 
    // swap the first and the third column - at the moment, the matrix contains
    // | z y x |
    int i;
    for (i = 0; i < 3; i++) {
        valib_swap_values(&Q[i], &Q[6+i]);
    }
 
    // verify that the matrix has a right-hand oriented coordinate system,
    // Q_1 x Q_2 = Q_3
    valib_cross_prod3(&Q[0], &Q[3], vec);
 
    // revert sign of the new x-axis to keep right-hand oriented coordinate
    // system
    factor = valib_scalar_prod3(vec, &Q[6]);
    if (factor < 0.) {
        for (i = 0; i < 3; i++) {
            Q[i] = -Q[i];
        }
    }
 
    // transpose the matrix
    valib_transpose3(Q);
}
 
/***************************************************************************/
#endif // VA_LINALG3_H