psi4-data 1:0.3-5 / usr / share / psi / python / qcdb / vecutil.py

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# PSI4: an ab initio quantum chemistry software package
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r"""File for accessory procedures in the chem module.
Credit for the libmints vector3 class to Justin M. Turney and
incremental improvements by other psi4 developers.

"""
import math
import copy
from exceptions import *

ZERO = 1.0E-14


def norm(v):
    """Compute the magnitude of vector *v*."""
    return math.sqrt(sum(v[i] * v[i] for i in range(len(v))))


def add(v, u):
    """Compute sum of vectors *v* and *u*."""
    return [u[i] + v[i] for i in range(len(v))]


def sub(v, u):
    """Compute difference of vectors *v* - *u*."""
    return [v[i] - u[i] for i in range(len(v))]


def dot(v, u):
    """Compute dot product of vectors *v* and *u*."""
    return sum(u[i] * v[i] for i in range(len(v)))


def scale(v, d):
    """Compute by-element scale by *d* of vector *v*."""
    return [d * v[i] for i in range(len(v))]


def naivemult(v, u):
    """Compute by-element multiplication of vectors *v* and *u*."""
    if len(u) != len(v):
        raise ValidationError('naivemult() only defined for vectors of same length \n')
    return [u[i] * v[i] for i in range(len(v))]


def normalize(v):
    """Compute normalized vector *v*."""
    vmag = norm(v)
    return [v[i] / vmag  for i in range(len(v))]


def distance(v, u):
    """Compute the distance between points defined by vectors *v* and *u*."""
    return norm(sub(v, u))


def cross(v, u):
    """Compute cross product of length 3 vectors *v* and *u*."""
    if len(u) != 3 or len(v) != 3:
        raise ValidationError('cross() only defined for vectors of length 3\n')
    return [v[1] * u[2] - v[2] * u[1],
            v[2] * u[0] - v[0] * u[2],
            v[0] * u[1] - v[1] * u[0]]


def rotate(v, theta, axis):
    """Rotate length 3 vector *v* about *axis* by *theta* radians."""
    if len(v) != 3 or len(axis) != 3:
        raise ValidationError('rotate() only defined for vectors of length 3\n')

    unitaxis = normalize(copy.deepcopy(axis))
    # split into parallel and perpendicular components along axis
    parallel = scale(axis, dot(v, axis) / dot(axis, axis))
    perpendicular = sub(v, parallel)
    # form unit vector perpendicular to parallel and perpendicular
    third_axis = perp_unit(axis, perpendicular)
    third_axis = scale(third_axis, norm(perpendicular))

    result = add(parallel, add(scale(perpendicular, math.cos(theta)), scale(third_axis, math.sin(theta))))
    for item in range(len(result)):
        if math.fabs(result[item]) < ZERO:
            result[item] = 0.0
    return result


def perp_unit(u, v):
    """Compute unit vector perpendicular to length 3 vectors *u* and *v*."""
    if len(u) != 3 or len(v) != 3:
        raise ValidationError('perp_unit() only defined for vectors of length 3\n')

    # try cross product
    result = cross(u, v)
    resultdotresult = dot(result, result)

    if resultdotresult < 1.E-16:
        # cross product is too small to normalize
        # find the largest of this and v
        dotprodt = dot(u, u)
        dotprodv = dot(v, v)
        if dotprodt < dotprodv:
            d = copy.deepcopy(v)
            dotprodd = dotprodv
        else:
            d = copy.deepcopy(u)
            dotprodd = dotprodt

        # see if d is big enough
        if dotprodd < 1.e-16:
            # choose an arbitrary vector, since the biggest vector is small
            result = [1.0, 0.0, 0.0]
            return result
        else:
            # choose a vector perpendicular to d
            # choose it in one of the planes xy, xz, yz
            # choose the plane to be that which contains the two largest components of d
            absd = [math.fabs(d[0]), math.fabs(d[1]), math.fabs(d[2])]
            if (absd[1] - absd[0]) > 1.0e-12:
            #if absd[0] < absd[1]:
                axis0 = 1
                if (absd[2] - absd[0]) > 1.0e-12:
                #if absd[0] < absd[2]:
                    axis1 = 2
                else:
                    axis1 = 0
            else:
                axis0 = 0
                if (absd[2] - absd[1]) > 1.0e-12:
                #if absd[1] < absd[2]:
                    axis1 = 2
                else:
                    axis1 = 1
            result = [0.0, 0.0, 0.0]
            # do the pi/2 rotation in the plane
            result[axis0] = d[axis1]
            result[axis1] = -1.0 * d[axis0]
        result = normalize(result)
        return result

    else:
        # normalize the cross product and return the result
        result = scale(result, 1.0 / math.sqrt(resultdotresult))
        return result


def determinant(mat):
    """Given 3x3 matrix *mat*, compute the determinat

    """
    if len(mat) != 3 or len(mat[0]) != 3 or len(mat[1]) != 3 or len(mat[2]) != 3:
        raise ValidationError('determinant() only defined for arrays of dimension 3x3\n')

    det = mat[0][0] * mat[1][1] * mat[2][2] - mat[0][2] * mat[1][1] * mat[2][0] + \
          mat[0][1] * mat[1][2] * mat[2][0] - mat[0][1] * mat[1][0] * mat[2][2] + \
          mat[0][2] * mat[1][0] * mat[2][1] - mat[0][0] * mat[1][2] * mat[2][1]
    return det


def diagonalize3x3symmat(M):
    """Given an real symmetric 3x3 matrix *M*, compute the eigenvalues

    """
    if len(M) != 3 or len(M[0]) != 3 or len(M[1]) != 3 or len(M[2]) != 3:
        raise ValidationError('diagonalize3x3symmat() only defined for arrays of dimension 3x3\n')

    A = copy.deepcopy(M)  # Symmetric input matrix
    Q = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]  # Storage buffer for eigenvectors
    w = [A[0][0], A[1][1], A[2][2]]  # Storage buffer for eigenvalues
    # sd, so                  # Sums of diagonal resp. off-diagonal elements
    # s, c, t                 # sin(phi), cos(phi), tan(phi) and temporary storage
    # g, h, z, theta          # More temporary storage

    # Calculate SQR(tr(A))
    sd = 0.0
    for i in range(3):
        sd += math.fabs(w[i])
    sd = sd * sd

    # Main iteration loop
    for nIter in range(50):

        # Test for convergence
        so = 0.0
        for p in range(3):
            for q in range(p + 1, 3):
                so += math.fabs(A[p][q])
        if so == 0.0:
            return w, Q  # return eval, evec

        if nIter < 4:
            thresh = 0.2 * so / (3 * 3)
        else:
            thresh = 0.0

        # Do sweep
        for p in range(3):
            for q in range(p + 1, 3):

                g = 100.0 * math.fabs(A[p][q])
                if nIter > 4  and (math.fabs(w[p]) + g == math.fabs(w[p])) and \
                    (math.fabs(w[q]) + g == math.fabs(w[q])):
                    A[p][q] = 0.0

                elif math.fabs(A[p][q]) > thresh:

                    # Calculate Jacobi transformation
                    h = w[q] - w[p]
                    if math.fabs(h) + g == math.fabs(h):
                        t = A[p][q] / h
                    else:
                        theta = 0.5 * h / A[p][q]
                        if theta < 0.0:
                            t = -1.0 / (math.sqrt(1.0 + theta * theta) - theta)
                        else:
                            t = 1.0 / (math.sqrt(1.0 + theta * theta) + theta)

                    c = 1.0 / math.sqrt(1.0 + t * t)
                    s = t * c
                    z = t * A[p][q]

                    # Apply Jacobi transformation
                    A[p][q] = 0.0
                    w[p] -= z
                    w[q] += z

                    for r in range(p):
                        t = A[r][p]
                        A[r][p] = c * t - s * A[r][q]
                        A[r][q] = s * t + c * A[r][q]

                    for r in range(p + 1, q):
                        t = A[p][r]
                        A[p][r] = c * t - s * A[r][q]
                        A[r][q] = s * t + c * A[r][q]

                    for r in range(q + 1, 3):
                        t = A[p][r]
                        A[p][r] = c * t - s * A[q][r]
                        A[q][r] = s * t + c * A[q][r]

                    # Update eigenvectors
                    for r in range(3):
                        t = Q[r][p]
                        Q[r][p] = c * t - s * Q[r][q]
                        Q[r][q] = s * t + c * Q[r][q]

    return None


def zero(m, n):
    """ Create zero matrix"""
    new_matrix = [[0 for row in range(n)] for col in range(m)]
    return new_matrix


def identity(m):
    """Create identity matrix"""
    new_matrix = zero(m, m)
    for i in range(m):
        new_matrix[i][i] = 1.0
    return new_matrix


def show(matrix):
    """ Print out matrix"""
    for col in matrix:
        print col


def mscale(matrix, d):
    """Return *matrix* scaled by scalar *d*"""
    for i in range(len(matrix)):
        for j in range(len(matrix[0])):
            matrix[i][j] *= d
    return matrix


def mult(matrix1, matrix2):
    """ Matrix multiplication"""
    if len(matrix1[0]) != len(matrix2):
        # Check matrix dimensions
        raise ValidationError('Matrices must be m*n and n*p to multiply!')

    else:
        # Multiply if correct dimensions
        try:
            new_matrix = zero(len(matrix1), len(matrix2[0]))
            for i in range(len(matrix1)):
                for j in range(len(matrix2[0])):
                    for k in range(len(matrix2)):
                        new_matrix[i][j] += matrix1[i][k] * matrix2[k][j]
        except TypeError:
            new_matrix = zero(len(matrix1), 1)
            for i in range(len(matrix1)):
                for k in range(len(matrix2)):
                    new_matrix[i][0] += matrix1[i][k] * matrix2[k]
        return new_matrix


def transpose(matrix):
    """Return matrix transpose"""
    if len(matrix[0]) != len(matrix):
        # Check matrix dimensions
        raise ValidationError('Matrices must be square.')

    tmat = [list(i) for i in zip(*matrix)]
    return tmat


def matadd(matrix1, matrix2, fac1=1.0, fac2=1.0):
    """Matrix addition"""
    if (len(matrix1[0]) != len(matrix2[0])) or (len(matrix1) != len(matrix2)):
        raise ValidationError('Matrices must be same dimension to add.')
    new_matrix = zero(len(matrix1), len(matrix1[0]))
    for i in range(len(matrix1)):
        for j in range(len(matrix1[0])):
            new_matrix[i][j] = fac1 * matrix1[i][j] + fac2 * matrix2[i][j]
    return new_matrix