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polyeval.pyx
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# -*- coding: utf-8 -*-
#
# WLSQM (Weighted Least SQuares Meshless): a fast and accurate meshless least-squares interpolator for Python, for scalar-valued data defined as point values on 1D, 2D and 3D point clouds.
#
# Evaluation of Taylor expansions and general polynomials up to 4th order in 1D, 2D and 3D.
#
# JJ 2016-11-30
# Set Cython compiler directives. This section must appear before any code!
#
# For available directives, see:
#
# http://docs.cython.org/en/latest/src/reference/compilation.html
#
# cython: wraparound = False
# cython: boundscheck = False
# cython: cdivision = True
from __future__ import division, print_function, absolute_import
# BUG in Cython 0.20.1post0: /usr/lib/python2.7/dist-packages/Cython/Includes/libc/math.pxd
# defines fma(double x, double y), but it should be fma(double x, double y, double z)
# so we import it manually.
#
#from libc.math cimport fma # this works in newer Cythons, the bug has been fixed
cdef extern from "<math.h>" nogil:
double fma(double x, double y, double z)
cimport wlsqm.fitter.defs as defs # C constants
####################################################
# Polynomial evaluation
####################################################
# These are implemented as a separate layer in order to allow us to re-use the same implementation
# for interpolating both the function value and the derivatives of the model.
# Evaluate an up to 4th order Taylor series expansion in 3D space, with its origin at (xi,yi,zi).
#
# This version uses "partially baked" coefficients, already accounting for the constant factors in the Taylor series!
# (This allows expressing the derivative information at the point xi in the raw coefficient data.)
#
# The implementation uses a symmetric Horner-like form with fused multiply-adds.
# (Note however that some of the symmetry is lost due to the way the partial results are summed.)
#
# order : in, the order of the expansion (0,1,2,3 or 4)
# fi : in, coefficient array ("order" determines the number of entries, no separate size parameter needed)
# The ordering of the coefficients follows the numbering of the i3_* constants in wlsqm.fitter.defs.
# Here fi[ i3_F_c ] is f at xi, fi[ i3_X ] is df/fx at xi, fi[ i3_Y ] is df/dy at xi, fi[ i3_X2 ] is d2f/dx2 at xi, ...
# xi : in, origin of the model, x component
# yi : in, origin of the model, y component
# zi : in, origin of the model, z component
# x : in, the points where to evaluate the Taylor expansion
# out : out, array of size (x.shape[0],); the result
#
# Return value: 0 on success, anything else indicates failure
#
cdef int taylor_3D( int order, double* fi, double xi, double yi, double zi, double[::view.generic,::view.contiguous] x, double* out ) nogil:
DEF onesixth = 1./6.
DEF one24th = 1./24.
if order not in [0,1,2,3,4]:
return -1
# with gil:
# raise ValueError( "order must be 1, 2, 3 or 4; got %d" % order )
cdef int n = x.shape[0]
cdef int k
cdef double dx, dy, dz
cdef double dxdy, dydz, dxdz
cdef double acc1, acc2 # accumulators
cdef double resX, resY, resZ, resXY, resYZ, resXZ, resXYZ # intermediate results
if order == 4:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
dxdy = dx*dy
dydz = dy*dz
dxdz = dx*dz
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# dy2 = dy*dy
# dz2 = dz*dz
# dx3 = dx2*dx
# dy3 = dy2*dy
# dz3 = dz2*dz
# dxdy = dx*dy
# dydz = dy*dz
# dxdz = dx*dz
# out[k] = fi[ i3_F_c ] + dx*fi[ i3_X_c ] + dy*fi[ i3_Y_c ] + dz*fi[ i3_Z_c ] \
#
# + 0.5*dx2*fi[ i3_X2_c ] + dxdy*fi[ i3_XY_c ] + 0.5*dy2*fi[ i3_Y2_c ] + dydz*fi[ i3_YZ_c ] + 0.5*dz2*fi[ i3_Z2_c ] + dxdz*fi[ i3_XZ_c ] \
#
# + onesixth*dx3*fi[ i3_X3_c ] + 0.5*dx2*dy*fi[ i3_X2Y_c ] + 0.5*dx*dy2*fi[ i3_XY2_c ] + onesixth*dy3*fi[ i3_Y3_c ] \
# + 0.5*dy2*dz*fi[ i3_Y2Z_c ] + 0.5*dy*dz2*fi[ i3_YZ2_c ] + onesixth*dz3*fi[ i3_Z3_c ] + 0.5*dx*dz2*fi[ i3_XZ2_c ] + 0.5*dx2*dz*fi[ i3_X2Z_c ] + dx*dy*dz*fi[ i3_XYZ_c ] \
#
# + one24th*dx2*dx2*fi[ i3_X4_c ] + onesixth*dx3*dy*fi[ i3_X3Y_c ] + 0.25*dxdy*dxdy*fi[ i3_X2Y2_c ] + onesixth*dx*dy3*fi[ i3_XY3_c ] + one24th*dy2*dy2*fi[ i3_Y4_c ] \
# + onesixth*dy3*dz*fi[ i3_Y3Z_c ] + 0.25*dydz*dydz*fi[ i3_Y2Z2_c ] + onesixth*dy*dz3*fi[ i3_YZ3_c ] + one24th*dz2*dz2*fi[ i3_Z4_c ] + onesixth*dx*dz3*fi[ i3_XZ3_c ] \
# + 0.25*dxdz*dxdz*fi[ i3_X2Z2_c ] + onesixth*dx3*dz*fi[ i3_X3Z_c ] + 0.5*dx2*dydz*fi[ i3_X2YZ_c ] + 0.5*dy2*dxdz*fi[ i3_XY2Z_c ] + 0.5*dxdy*dz2*fi[ i3_XYZ2_c ]
#
# # symmetric Horner-like form:
# out[k] = fi[ i3_F_c ] + dx*( fi[ i3_X_c ] + dx*( 0.5*( fi[ i3_X2_c ] + dy*fi[ i3_X2Y_c ] + dz*fi[ i3_X2Z_c ] + dydz*fi[ i3_X2YZ_c ] )
# + dx*( onesixth*( fi[ i3_X3_c ] + dy*fi[ i3_X3Y_c ] + dz*fi[ i3_X3Z_c ] )
# + dx*( one24th*fi[ i3_X4_c ] ) ) ) ) \
#
# + dy*( fi[ i3_Y_c ] + dy*( 0.5*( fi[ i3_Y2_c ] + dx*fi[ i3_XY2_c ] + dz*fi[ i3_Y2Z_c ] + dxdz*fi[ i3_XY2Z_c ] )
# + dy*( onesixth*( fi[ i3_Y3_c ] + dx*fi[ i3_XY3_c ] + dz*fi[ i3_Y3Z_c ] )
# + dy*( one24th*fi[ i3_Y4_c ] ) ) ) ) \
#
# + dz*( fi[ i3_Z_c ] + dz*( 0.5*( fi[ i3_Z2_c ] + dx*fi[ i3_XZ2_c ] + dy*fi[ i3_YZ2_c ] + dxdy*fi[ i3_XYZ2_c ] )
# + dz*( onesixth*( fi[ i3_Z3_c ] + dx*fi[ i3_XZ3_c ] + dy*fi[ i3_YZ3_c ] )
# + dz*( one24th*fi[ i3_Z4_c ] ) ) ) ) \
#
# + dxdy*( fi[ i3_XY_c ] + dxdy*0.25*fi[ i3_X2Y2_c ] ) \
# + dydz*( fi[ i3_YZ_c ] + dydz*0.25*fi[ i3_Y2Z2_c ] ) \
# + dxdz*( fi[ i3_XZ_c ] + dxdz*0.25*fi[ i3_X2Z2_c ] ) \
#
# + dx*dy*dz*fi[ i3_XYZ_c ]
# symmetric Horner-like form using fused multiply-add (fma):
# "X" terms (the "..." in dx*... above)
acc1 = fma( dy, fi[ defs.i3_X3Y_c ], fi[ defs.i3_X3_c ] )
acc1 = fma( dz, fi[ defs.i3_X3Z_c ], acc1 )
acc1 *= onesixth
acc1 = fma( dx, one24th*fi[ defs.i3_X4_c ], acc1 )
acc2 = fma( dy, fi[ defs.i3_X2Y_c ], fi[ defs.i3_X2_c ] )
acc2 = fma( dz, fi[ defs.i3_X2Z_c ], acc2 )
acc2 = fma( dydz, fi[ defs.i3_X2YZ_c ], acc2 )
acc2 *= 0.5
acc2 = fma( dx, acc1, acc2 )
resX = fma( dx, acc2, fi[ defs.i3_X_c ] )
# "Y" terms
acc1 = fma( dx, fi[ defs.i3_XY3_c ], fi[ defs.i3_Y3_c ] )
acc1 = fma( dz, fi[ defs.i3_Y3Z_c ], acc1 )
acc1 *= onesixth
acc1 = fma( dy, one24th*fi[ defs.i3_Y4_c ], acc1 )
acc2 = fma( dx, fi[ defs.i3_XY2_c ], fi[ defs.i3_Y2_c ] )
acc2 = fma( dz, fi[ defs.i3_Y2Z_c ], acc2 )
acc2 = fma( dxdz, fi[ defs.i3_XY2Z_c ], acc2 )
acc2 *= 0.5
acc2 = fma( dy, acc1, acc2 )
resY = fma( dy, acc2, fi[ defs.i3_Y_c ] )
# "Z" terms
acc1 = fma( dx, fi[ defs.i3_XZ3_c ], fi[ defs.i3_Z3_c ] )
acc1 = fma( dy, fi[ defs.i3_YZ3_c ], acc1 )
acc1 *= onesixth
acc1 = fma( dz, one24th*fi[ defs.i3_Z4_c ], acc1 )
acc2 = fma( dx, fi[ defs.i3_XZ2_c ], fi[ defs.i3_Z2_c ] )
acc2 = fma( dy, fi[ defs.i3_YZ2_c ], acc2 )
acc2 = fma( dxdy, fi[ defs.i3_XYZ2_c ], acc2 )
acc2 *= 0.5
acc2 = fma( dz, acc1, acc2 )
resZ = fma( dz, acc2, fi[ defs.i3_Z_c ] )
# "XY", "YZ", "XZ" terms
resXY = fma( dxdy, 0.25*fi[ defs.i3_X2Y2_c ], fi[ defs.i3_XY_c ] )
resYZ = fma( dydz, 0.25*fi[ defs.i3_Y2Z2_c ], fi[ defs.i3_YZ_c ] )
resXZ = fma( dxdz, 0.25*fi[ defs.i3_X2Z2_c ], fi[ defs.i3_XZ_c ] )
# sum the contributions (in an approximately increasing order of magnitude)
#
acc1 = dx*dy*dz*fi[ defs.i3_XYZ_c ]
acc1 = fma( dxdy, resXY, acc1 )
acc1 = fma( dydz, resYZ, acc1 )
acc1 = fma( dxdz, resXZ, acc1 )
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 = fma( dz, resZ, acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
elif order == 3:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
dxdy = dx*dy
dydz = dy*dz
dxdz = dx*dz
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# dy2 = dy*dy
# dz2 = dz*dz
# dx3 = dx2*dx
# dy3 = dy2*dy
# dz3 = dz2*dz
# dxdy = dx*dy
# dydz = dy*dz
# dxdz = dx*dz
# out[k] = fi[ i3_F_c ] + dx*fi[ i3_X_c ] + dy*fi[ i3_Y_c ] + dz*fi[ i3_Z_c ] \
#
# + 0.5*dx2*fi[ i3_X2_c ] + dxdy*fi[ i3_XY_c ] + 0.5*dy2*fi[ i3_Y2_c ] + dydz*fi[ i3_YZ_c ] + 0.5*dz2*fi[ i3_Z2_c ] + dxdz*fi[ i3_XZ_c ] \
#
# + onesixth*dx3*fi[ i3_X3_c ] + 0.5*dx2*dy*fi[ i3_X2Y_c ] + 0.5*dx*dy2*fi[ i3_XY2_c ] + onesixth*dy3*fi[ i3_Y3_c ] \
# + 0.5*dy2*dz*fi[ i3_Y2Z_c ] + 0.5*dy*dz2*fi[ i3_YZ2_c ] + onesixth*dz3*fi[ i3_Z3_c ] + 0.5*dx*dz2*fi[ i3_XZ2_c ] + 0.5*dx2*dz*fi[ i3_X2Z_c ] + dx*dy*dz*fi[ i3_XYZ_c ]
#
# # symmetric Horner-like form:
# out[k] = fi[ i3_F_c ] + dx*( fi[ i3_X_c ] + dx*( 0.5*( fi[ i3_X2_c ] + dy*fi[ i3_X2Y_c ] + dz*fi[ i3_X2Z_c ] )
# + dx*( onesixth*( fi[ i3_X3_c ] ) ) ) ) \
#
# + dy*( fi[ i3_Y_c ] + dy*( 0.5*( fi[ i3_Y2_c ] + dx*fi[ i3_XY2_c ] + dz*fi[ i3_Y2Z_c ] )
# + dy*( onesixth*( fi[ i3_Y3_c ] ) ) ) ) \
#
# + dz*( fi[ i3_Z_c ] + dz*( 0.5*( fi[ i3_Z2_c ] + dx*fi[ i3_XZ2_c ] + dy*fi[ i3_YZ2_c ] )
# + dz*( onesixth*( fi[ i3_Z3_c ] ) ) ) ) \
#
# + dxdy*( fi[ i3_XY_c ] ) \
# + dydz*( fi[ i3_YZ_c ] ) \
# + dxdz*( fi[ i3_XZ_c ] ) \
#
# + dx*dy*dz*fi[ i3_XYZ_c ]
# "X" terms (the "..." in dx*... above)
acc2 = fma( dy, fi[ defs.i3_X2Y_c ], fi[ defs.i3_X2_c ] )
acc2 = fma( dz, fi[ defs.i3_X2Z_c ], acc2 )
acc2 *= 0.5
acc2 = fma( dx, onesixth*fi[ defs.i3_X3_c ], acc2 )
resX = fma( dx, acc2, fi[ defs.i3_X_c ] )
# "Y" terms
acc2 = fma( dx, fi[ defs.i3_XY2_c ], fi[ defs.i3_Y2_c ] )
acc2 = fma( dz, fi[ defs.i3_Y2Z_c ], acc2 )
acc2 *= 0.5
acc2 = fma( dy, onesixth*fi[ defs.i3_Y3_c ], acc2 )
resY = fma( dy, acc2, fi[ defs.i3_Y_c ] )
# "Z" terms
acc2 = fma( dx, fi[ defs.i3_XZ2_c ], fi[ defs.i3_Z2_c ] )
acc2 = fma( dy, fi[ defs.i3_YZ2_c ], acc2 )
acc2 *= 0.5
acc2 = fma( dz, onesixth*fi[ defs.i3_Z3_c ], acc2 )
resZ = fma( dz, acc2, fi[ defs.i3_Z_c ] )
# sum the contributions
#
acc1 = dx*dy*dz*fi[ defs.i3_XYZ_c ]
acc1 = fma( dxdy, fi[ defs.i3_XY_c ], acc1 )
acc1 = fma( dydz, fi[ defs.i3_YZ_c ], acc1 )
acc1 = fma( dxdz, fi[ defs.i3_XZ_c ], acc1 )
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 = fma( dz, resZ, acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
elif order == 2:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
dxdy = dx*dy
dydz = dy*dz
dxdz = dx*dz
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# dy2 = dy*dy
# dz2 = dz*dz
# dxdy = dx*dy
# dydz = dy*dz
# dxdz = dx*dz
# out[k] = fi[ i3_F_c ] + dx*fi[ i3_X_c ] + dy*fi[ i3_Y_c ] + dz*fi[ i3_Z_c ] \
# + 0.5*dx2*fi[ i3_X2_c ] + dxdy*fi[ i3_XY_c ] + 0.5*dy2*fi[ i3_Y2_c ] + dydz*fi[ i3_YZ_c ] + 0.5*dz2*fi[ i3_Z2_c ] + dxdz*fi[ i3_XZ_c ]
#
# # symmetric Horner-like form:
# out[k] = fi[ i3_F_c ] + dx*( fi[ i3_X_c ] + dx*( 0.5*( fi[ i3_X2_c ] ) ) ) \
# + dy*( fi[ i3_Y_c ] + dy*( 0.5*( fi[ i3_Y2_c ] ) ) ) \
# + dz*( fi[ i3_Z_c ] + dz*( 0.5*( fi[ i3_Z2_c ] ) ) ) \
#
# + dxdy*( fi[ i3_XY_c ] ) \
# + dydz*( fi[ i3_YZ_c ] ) \
# + dxdz*( fi[ i3_XZ_c ] )
resX = fma( dx, 0.5*fi[ defs.i3_X2_c ], fi[ defs.i3_X_c ] )
resY = fma( dy, 0.5*fi[ defs.i3_Y2_c ], fi[ defs.i3_Y_c ] )
resZ = fma( dz, 0.5*fi[ defs.i3_Z2_c ], fi[ defs.i3_Z_c ] )
# sum the contributions
#
acc1 = dxdy*fi[ defs.i3_XY_c ]
acc1 = fma( dydz, fi[ defs.i3_YZ_c ], acc1 )
acc1 = fma( dxdz, fi[ defs.i3_XZ_c ], acc1 )
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 = fma( dz, resZ, acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
elif order == 1:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
# # for documentation only:
# out[k] = fi[ i3_F_c ] + dx*fi[ i3_X_c ] + dy*fi[ i3_Y_c ] + dz*fi[ i3_Z_c ]
# sum the contributions
#
acc1 = dx*fi[ defs.i3_X_c ]
acc1 = fma( dy, fi[ defs.i3_Y_c ], acc1 )
acc1 = fma( dz, fi[ defs.i3_Z_c ], acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
else: # order == 0:
for k in range(n):
out[k] = fi[ defs.i3_F_c ]
return 0
# Same strategy as above, but for a generic polynomial (no partially baked coefficients).
#
# Now fi[ i3_F_c ] is the constant term, fi[ i3_X ] is the coefficient of (x - xi), fi[ i3_Y ] is the coefficient of (y - yi), fi[ i3_X2 ] is the coefficient of (x - xi)**2, ...
#
cdef int general_3D( int order, double* fi, double xi, double yi, double zi, double[::view.generic,::view.contiguous] x, double* out ) nogil:
if order not in [0,1,2,3,4]:
return -1
# with gil:
# raise ValueError( "order must be 1, 2, 3 or 4; got %d" % order )
cdef int n = x.shape[0]
cdef int k
cdef double dx, dy, dz
cdef double dxdy, dydz, dxdz
cdef double acc1, acc2 # accumulators
cdef double resX, resY, resZ, resXY, resYZ, resXZ, resXYZ # intermediate results
if order == 4:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
dxdy = dx*dy
dydz = dy*dz
dxdz = dx*dz
# symmetric Horner-like form using fused multiply-add (fma):
# "X" terms
acc1 = fma( dy, fi[ defs.i3_X3Y_c ], fi[ defs.i3_X3_c ] )
acc1 = fma( dz, fi[ defs.i3_X3Z_c ], acc1 )
acc1 = fma( dx, fi[ defs.i3_X4_c ], acc1 )
acc2 = fma( dy, fi[ defs.i3_X2Y_c ], fi[ defs.i3_X2_c ] )
acc2 = fma( dz, fi[ defs.i3_X2Z_c ], acc2 )
acc2 = fma( dydz, fi[ defs.i3_X2YZ_c ], acc2 )
acc2 = fma( dx, acc1, acc2 )
resX = fma( dx, acc2, fi[ defs.i3_X_c ] )
# "Y" terms
acc1 = fma( dx, fi[ defs.i3_XY3_c ], fi[ defs.i3_Y3_c ] )
acc1 = fma( dz, fi[ defs.i3_Y3Z_c ], acc1 )
acc1 = fma( dy, fi[ defs.i3_Y4_c ], acc1 )
acc2 = fma( dx, fi[ defs.i3_XY2_c ], fi[ defs.i3_Y2_c ] )
acc2 = fma( dz, fi[ defs.i3_Y2Z_c ], acc2 )
acc2 = fma( dxdz, fi[ defs.i3_XY2Z_c ], acc2 )
acc2 = fma( dy, acc1, acc2 )
resY = fma( dy, acc2, fi[ defs.i3_Y_c ] )
# "Z" terms
acc1 = fma( dx, fi[ defs.i3_XZ3_c ], fi[ defs.i3_Z3_c ] )
acc1 = fma( dy, fi[ defs.i3_YZ3_c ], acc1 )
acc1 = fma( dz, fi[ defs.i3_Z4_c ], acc1 )
acc2 = fma( dx, fi[ defs.i3_XZ2_c ], fi[ defs.i3_Z2_c ] )
acc2 = fma( dy, fi[ defs.i3_YZ2_c ], acc2 )
acc2 = fma( dxdy, fi[ defs.i3_XYZ2_c ], acc2 )
acc2 = fma( dz, acc1, acc2 )
resZ = fma( dz, acc2, fi[ defs.i3_Z_c ] )
# "XY", "YZ", "XZ" terms
resXY = fma( dxdy, fi[ defs.i3_X2Y2_c ], fi[ defs.i3_XY_c ] )
resYZ = fma( dydz, fi[ defs.i3_Y2Z2_c ], fi[ defs.i3_YZ_c ] )
resXZ = fma( dxdz, fi[ defs.i3_X2Z2_c ], fi[ defs.i3_XZ_c ] )
# sum the contributions (in an approximately increasing order of magnitude)
#
acc1 = dx*dy*dz*fi[ defs.i3_XYZ_c ]
acc1 = fma( dxdy, resXY, acc1 )
acc1 = fma( dydz, resYZ, acc1 )
acc1 = fma( dxdz, resXZ, acc1 )
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 = fma( dz, resZ, acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
elif order == 3:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
dxdy = dx*dy
dydz = dy*dz
dxdz = dx*dz
# "X" terms
acc2 = fma( dy, fi[ defs.i3_X2Y_c ], fi[ defs.i3_X2_c ] )
acc2 = fma( dz, fi[ defs.i3_X2Z_c ], acc2 )
acc2 = fma( dx, fi[ defs.i3_X3_c ], acc2 )
resX = fma( dx, acc2, fi[ defs.i3_X_c ] )
# "Y" terms
acc2 = fma( dx, fi[ defs.i3_XY2_c ], fi[ defs.i3_Y2_c ] )
acc2 = fma( dz, fi[ defs.i3_Y2Z_c ], acc2 )
acc2 = fma( dy, fi[ defs.i3_Y3_c ], acc2 )
resY = fma( dy, acc2, fi[ defs.i3_Y_c ] )
# "Z" terms
acc2 = fma( dx, fi[ defs.i3_XZ2_c ], fi[ defs.i3_Z2_c ] )
acc2 = fma( dy, fi[ defs.i3_YZ2_c ], acc2 )
acc2 = fma( dz, fi[ defs.i3_Z3_c ], acc2 )
resZ = fma( dz, acc2, fi[ defs.i3_Z_c ] )
# sum the contributions
#
acc1 = dx*dy*dz*fi[ defs.i3_XYZ_c ]
acc1 = fma( dxdy, fi[ defs.i3_XY_c ], acc1 )
acc1 = fma( dydz, fi[ defs.i3_YZ_c ], acc1 )
acc1 = fma( dxdz, fi[ defs.i3_XZ_c ], acc1 )
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 = fma( dz, resZ, acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
elif order == 2:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
dxdy = dx*dy
dydz = dy*dz
dxdz = dx*dz
resX = fma( dx, fi[ defs.i3_X2_c ], fi[ defs.i3_X_c ] )
resY = fma( dy, fi[ defs.i3_Y2_c ], fi[ defs.i3_Y_c ] )
resZ = fma( dz, fi[ defs.i3_Z2_c ], fi[ defs.i3_Z_c ] )
# sum the contributions
#
acc1 = dxdy*fi[ defs.i3_XY_c ]
acc1 = fma( dydz, fi[ defs.i3_YZ_c ], acc1 )
acc1 = fma( dxdz, fi[ defs.i3_XZ_c ], acc1 )
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 = fma( dz, resZ, acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
elif order == 1:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dz = x[k,2] - zi
# sum the contributions
#
acc1 = dx*fi[ defs.i3_X_c ]
acc1 = fma( dy, fi[ defs.i3_Y_c ], acc1 )
acc1 = fma( dz, fi[ defs.i3_Z_c ], acc1 )
acc1 += fi[ defs.i3_F_c ]
out[k] = acc1
else: # order == 0:
for k in range(n):
out[k] = fi[ defs.i3_F_c ]
return 0
# Evaluate an up to 4th order Taylor series expansion in the plane, with its origin at (xi,yi).
#
# This version uses "partially baked" coefficients, already accounting for the constant factors in the Taylor series!
# (This allows expressing the derivative information at the point xi in the raw coefficient data.)
#
# The implementation uses a symmetric Horner-like form with fused multiply-adds.
# (Note however that some of the symmetry is lost due to the way the partial results are summed.)
#
# order : in, the order of the expansion (0,1,2,3 or 4)
# fi : in, coefficient array ("order" determines the number of entries, no separate size parameter needed)
# The ordering of the coefficients follows the numbering of the i2_* constants in wlsqm.fitter.defs.
# Here fi[ i2_F_c ] is f at xi, fi[ i2_X ] is df/fx at xi, fi[ i2_Y ] is df/dy at xi, fi[ i2_X2 ] is d2f/dx2 at xi, ...
# xi : in, origin of the model, x component
# yi : in, origin of the model, y component
# x : in, the points where to evaluate the Taylor expansion
# out : out, array of size (x.shape[0],); the result
#
# Return value: 0 on success, anything else indicates failure
#
cdef int taylor_2D( int order, double* fi, double xi, double yi, double[::view.generic,::view.contiguous] x, double* out ) nogil:
DEF onesixth = 1./6.
DEF one24th = 1./24.
if order not in [0,1,2,3,4]:
return -1
# with gil:
# raise ValueError( "order must be 1, 2, 3 or 4; got %d" % order )
cdef int n = x.shape[0]
cdef int k
cdef double dx, dy
cdef double dxdy
cdef double acc1, acc2 # accumulators
cdef double resX, resY, resXY # intermediate results
if order == 4:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dxdy = dx*dy
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# dy2 = dy*dy
# dx3 = dx2*dx
# dy3 = dy2*dy
# dxdy = dx*dy
# out[k] = fi[ i2_F_c ] + dx*fi[ i2_X_c ] + dy*fi[ i2_Y_c ] \
#
# + 0.5*dx2*fi[ i2_X2_c ] + dxdy*fi[ i2_XY_c ] + 0.5*dy2*fi[ i2_Y2_c ] \
#
# + onesixth*dx3*fi[ i2_X3_c ] + 0.5*dx2*dy*fi[ i2_X2Y_c ] + 0.5*dx*dy2*fi[ i2_XY2_c ] + onesixth*dy3*fi[ i2_Y3_c ] \
#
# + one24th*dx2*dx2*fi[ i2_X4_c ] + onesixth*dx3*dy*fi[ i2_X3Y_c ] + 0.25*dxdy*dxdy*fi[ i2_X2Y2_c ] + onesixth*dx*dy3*fi[ i2_XY3_c ] + one24th*dy2*dy2*fi[ i2_Y4_c ]
#
# # symmetric Horner-like form:
# out[k] = fi[ i2_F_c ] + dx*( fi[ i2_X_c ] + dx*( 0.5*( fi[ i2_X2_c ] + dy*fi[ i2_X2Y_c ] )
# + dx*( onesixth*( fi[ i2_X3_c ] + dy*fi[ i2_X3Y_c ] )
# + dx*( one24th*fi[ i2_X4_c ] ) ) ) ) \
#
# + dy*( fi[ i2_Y_c ] + dy*( 0.5*( fi[ i2_Y2_c ] + dx*fi[ i2_XY2_c ] )
# + dy*( onesixth*( fi[ i2_Y3_c ] + dx*fi[ i2_XY3_c ] )
# + dy*( one24th*fi[ i2_Y4_c ] ) ) ) ) \
#
# + dxdy*( fi[ i2_XY_c ] + dxdy*0.25*fi[ i2_X2Y2_c ] )
# symmetric Horner-like form using fused multiply-add (fma):
# "X" terms (the "..." in dx*... above)
acc1 = fma( dy, fi[ defs.i2_X3Y_c ], fi[ defs.i2_X3_c ] )
acc1 *= onesixth
acc1 = fma( dx, one24th*fi[ defs.i2_X4_c ], acc1 )
acc2 = fma( dy, fi[ defs.i2_X2Y_c ], fi[ defs.i2_X2_c ] )
acc2 *= 0.5
acc2 = fma( dx, acc1, acc2 )
resX = fma( dx, acc2, fi[ defs.i2_X_c ] )
# "Y" terms
acc1 = fma( dx, fi[ defs.i2_XY3_c ], fi[ defs.i2_Y3_c ] )
acc1 *= onesixth
acc1 = fma( dy, one24th*fi[ defs.i2_Y4_c ], acc1 )
acc2 = fma( dx, fi[ defs.i2_XY2_c ], fi[ defs.i2_Y2_c ] )
acc2 *= 0.5
acc2 = fma( dy, acc1, acc2 )
resY = fma( dy, acc2, fi[ defs.i2_Y_c ] )
# "XY" terms
resXY = fma( dxdy, 0.25*fi[ defs.i2_X2Y2_c ], fi[ defs.i2_XY_c ] )
# sum the contributions (in an approximately increasing order of magnitude)
#
acc1 = dxdy*resXY
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
elif order == 3:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dxdy = dx*dy
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# dy2 = dy*dy
# dx3 = dx2*dx
# dy3 = dy2*dy
# dxdy = dx*dy
# out[k] = fi[ i2_F_c ] + dx*fi[ i2_X_c ] + dy*fi[ i2_Y_c ] \
#
# + 0.5*dx2*fi[ i2_X2_c ] + dxdy*fi[ i2_XY_c ] + 0.5*dy2*fi[ i2_Y2_c ] \
#
# + onesixth*dx3*fi[ i2_X3_c ] + 0.5*dx2*dy*fi[ i2_X2Y_c ] + 0.5*dx*dy2*fi[ i2_XY2_c ] + onesixth*dy3*fi[ i2_Y3_c ]
#
# # symmetric Horner-like form:
# out[k] = fi[ i2_F_c ] + dx*( fi[ i2_X_c ] + dx*( 0.5*( fi[ i2_X2_c ] + dy*fi[ i2_X2Y_c ] )
# + dx*( onesixth*( fi[ i2_X3_c ] ) ) ) ) \
#
# + dy*( fi[ i2_Y_c ] + dy*( 0.5*( fi[ i2_Y2_c ] + dx*fi[ i2_XY2_c ] )
# + dy*( onesixth*( fi[ i2_Y3_c ] ) ) ) ) \
#
# + dxdy*( fi[ i2_XY_c ] )
# "X" terms (the "..." in dx*... above)
acc2 = fma( dy, fi[ defs.i2_X2Y_c ], fi[ defs.i2_X2_c ] )
acc2 *= 0.5
acc2 = fma( dx, onesixth*fi[ defs.i2_X3_c ], acc2 )
resX = fma( dx, acc2, fi[ defs.i2_X_c ] )
# "Y" terms
acc2 = fma( dx, fi[ defs.i2_XY2_c ], fi[ defs.i2_Y2_c ] )
acc2 *= 0.5
acc2 = fma( dy, onesixth*fi[ defs.i2_Y3_c ], acc2 )
resY = fma( dy, acc2, fi[ defs.i2_Y_c ] )
# sum the contributions
#
acc1 = dxdy*fi[ defs.i2_XY_c ]
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
elif order == 2:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dxdy = dx*dy
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# dy2 = dy*dy
# dxdy = dx*dy
# out[k] = fi[ i2_F_c ] + dx*fi[ i2_X_c ] + dy*fi[ i2_Y_c ] \
# + 0.5*dx2*fi[ i2_X2_c ] + dxdy*fi[ i2_XY_c ] + 0.5*dy2*fi[ i2_Y2_c ]
#
# # symmetric Horner-like form:
# out[k] = fi[ i2_F_c ] + dx*( fi[ i2_X_c ] + dx*( 0.5*( fi[ i2_X2_c ] ) ) ) \
# + dy*( fi[ i2_Y_c ] + dy*( 0.5*( fi[ i2_Y2_c ] ) ) ) \
#
# + dxdy*( fi[ i2_XY_c ] )
resX = fma( dx, 0.5*fi[ defs.i2_X2_c ], fi[ defs.i2_X_c ] )
resY = fma( dy, 0.5*fi[ defs.i2_Y2_c ], fi[ defs.i2_Y_c ] )
# sum the contributions
#
acc1 = dxdy*fi[ defs.i2_XY_c ]
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
elif order == 1:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
# # for documentation only:
# out[k] = fi[ i2_F_c ] + dx*fi[ i2_X_c ] + dy*fi[ i2_Y_c ]
# sum the contributions
#
acc1 = dx*fi[ defs.i2_X_c ]
acc1 = fma( dy, fi[ defs.i2_Y_c ], acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
else: # order == 0:
for k in range(n):
out[k] = fi[ defs.i2_F_c ]
return 0
# Same strategy as above, but for a generic polynomial (no partially baked coefficients).
#
# Now fi[ i2_F_c ] is the constant term, fi[ i2_X ] is the coefficient of (x - xi), fi[ i2_Y ] is the coefficient of (y - yi), fi[ i2_X2 ] is the coefficient of (x - xi)**2, ...
#
cdef int general_2D( int order, double* fi, double xi, double yi, double[::view.generic,::view.contiguous] x, double* out ) nogil:
if order not in [0,1,2,3,4]:
return -1
# with gil:
# raise ValueError( "order must be 1, 2, 3 or 4; got %d" % order )
cdef int n = x.shape[0]
cdef int k
cdef double dx, dy
cdef double dxdy
cdef double acc1, acc2 # accumulators
cdef double resX, resY, resXY # intermediate results
if order == 4:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dxdy = dx*dy
# symmetric Horner-like form using fused multiply-add (fma):
# "X" terms
acc1 = fma( dy, fi[ defs.i2_X3Y_c ], fi[ defs.i2_X3_c ] )
acc1 = fma( dx, fi[ defs.i2_X4_c ], acc1 )
acc2 = fma( dy, fi[ defs.i2_X2Y_c ], fi[ defs.i2_X2_c ] )
acc2 = fma( dx, acc1, acc2 )
resX = fma( dx, acc2, fi[ defs.i2_X_c ] )
# "Y" terms
acc1 = fma( dx, fi[ defs.i2_XY3_c ], fi[ defs.i2_Y3_c ] )
acc1 = fma( dy, fi[ defs.i2_Y4_c ], acc1 )
acc2 = fma( dx, fi[ defs.i2_XY2_c ], fi[ defs.i2_Y2_c ] )
acc2 = fma( dy, acc1, acc2 )
resY = fma( dy, acc2, fi[ defs.i2_Y_c ] )
# "XY" terms
resXY = fma( dxdy, fi[ defs.i2_X2Y2_c ], fi[ defs.i2_XY_c ] )
# sum the contributions
#
acc1 = dxdy*resXY
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
elif order == 3:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dxdy = dx*dy
# "X" terms
acc2 = fma( dy, fi[ defs.i2_X2Y_c ], fi[ defs.i2_X2_c ] )
acc2 = fma( dx, fi[ defs.i2_X3_c ], acc2 )
resX = fma( dx, acc2, fi[ defs.i2_X_c ] )
# "Y" terms
acc2 = fma( dx, fi[ defs.i2_XY2_c ], fi[ defs.i2_Y2_c ] )
acc2 = fma( dy, fi[ defs.i2_Y3_c ], acc2 )
resY = fma( dy, acc2, fi[ defs.i2_Y_c ] )
# sum the contributions
#
acc1 = dxdy*fi[ defs.i2_XY_c ]
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
elif order == 2:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
dxdy = dx*dy
resX = fma( dx, fi[ defs.i2_X2_c ], fi[ defs.i2_X_c ] )
resY = fma( dy, fi[ defs.i2_Y2_c ], fi[ defs.i2_Y_c ] )
# sum the contributions
#
acc1 = dxdy*fi[ defs.i2_XY_c ]
acc1 = fma( dx, resX, acc1 )
acc1 = fma( dy, resY, acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
elif order == 1:
for k in range(n):
dx = x[k,0] - xi
dy = x[k,1] - yi
# # for documentation only:
# out[k] = fi[ i2_F_c ] + dx*fi[ i2_X_c ] + dy*fi[ i2_Y_c ]
# sum the contributions
#
acc1 = dx*fi[ defs.i2_X_c ]
acc1 = fma( dy, fi[ defs.i2_Y_c ], acc1 )
acc1 += fi[ defs.i2_F_c ]
out[k] = acc1
else: # order == 0:
for k in range(n):
out[k] = fi[ defs.i2_F_c ]
return 0
# Evaluate an up to 4th order Taylor series expansion on the real axis, with its origin at (xi).
#
# The implementation uses Horner's form with fused multiply-adds.
#
# order : in, the order of the expansion (0,1,2,3 or 4)
# fi : in, coefficient array ("order" determines the number of entries, no separate size parameter needed)
# The ordering of the coefficients follows the numbering of the i1_* constants in wlsqm.fitter.defs.
# Here fi[ i1_F_c ] is f at xi, fi[ i1_X ] is df/dx at xi, fi[ i1_X2 ] is d2f/dx2 at xi, ...
# xi : in, origin of the model
# x : in, the points where to evaluate the Taylor expansion
# out : out, array of size (x.shape[0],); the result
#
# Return value: 0 on success, anything else indicates failure
#
cdef int taylor_1D( int order, double* fi, double xi, double[::view.generic] x, double* out ) nogil:
DEF onesixth = 1./6.
DEF one24th = 1./24.
if order not in [0,1,2,3,4]:
return -1
# with gil:
# raise ValueError( "order must be 1, 2, 3 or 4; got %d" % order )
cdef int n = x.shape[0]
cdef int k
cdef double dx
cdef double acc
if order == 4:
for k in range(n):
dx = x[k] - xi
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# out[k] = fi[ i1_F_c ] + dx*fi[ i1_X_c ] + 0.5*dx2*fi[ i1_X2_c ] + onesixth*dx2*dx*fi[ i1_X3_c ] + one24th*dx2*dx2*fi[ i1_X4_c ]
#
# # Horner form:
# out[k] = fi[ i1_F_c ] + dx*( fi[ i1_X_c ] + dx*( 0.5*fi[ i1_X2_c ] + dx*( onesixth*fi[ i1_X3_c ] + dx*( one24th*fi[ i1_X4_c ] ) ) ) )
# Horner form using fused multiply-add:
acc = fma( dx, one24th*fi[ defs.i1_X4_c ], onesixth*fi[ defs.i1_X3_c ] )
acc = fma( dx, acc, 0.5*fi[ defs.i1_X2_c ] )
acc = fma( dx, acc, fi[ defs.i1_X_c ] )
out[k] = fma( dx, acc, fi[ defs.i1_F_c ] )
elif order == 3:
for k in range(n):
dx = x[k] - xi
# # for documentation only:
#
# # naive form:
# dx2 = dx*dx
# out[k] = fi[ i1_F_c ] + dx*fi[ i1_X_c ] + 0.5*dx2*fi[ i1_X2_c ] + onesixth*dx2*dx*fi[ i1_X3_c ]
#
# # Horner form:
# out[k] = fi[ i1_F_c ] + dx*( fi[ i1_X_c ] + dx*( 0.5*fi[ i1_X2_c ] + dx*( onesixth*fi[ i1_X3_c ] ) ) )
# Horner form using fused multiply-add:
acc = fma( dx, onesixth*fi[ defs.i1_X3_c ], 0.5*fi[ defs.i1_X2_c ] )
acc = fma( dx, acc, fi[ defs.i1_X_c ] )
out[k] = fma( dx, acc, fi[ defs.i1_F_c ] )
elif order == 2:
for k in range(n):
dx = x[k] - xi
# # for documentation only:
#
# # naive form:
# out[k] = fi[ i1_F_c ] + dx*fi[ i1_X_c ] + 0.5*dx*dx*fi[ i1_X2_c ]
#
# # Horner form:
# out[k] = fi[ i1_F_c ] + dx*( fi[ i1_X_c ] + dx*( 0.5*fi[ i1_X2_c ] ) )
# Horner form using fused multiply-add:
acc = fma( dx, 0.5*fi[ defs.i1_X2_c ], fi[ defs.i1_X_c ] )
out[k] = fma( dx, acc, fi[ defs.i1_F_c ] )
elif order == 1:
for k in range(n):
dx = x[k] - xi
# out[k] = fi[ i1_F_c ] + dx*fi[ i1_X_c ]
# Horner form using fused multiply-add:
out[k] = fma( dx, fi[ defs.i1_X_c ], fi[ defs.i1_F_c ] ) # fused multiply-add, op1*op2 + op3
else: # order == 0:
for k in range(n):
out[k] = fi[ defs.i1_F_c ]
return 0
# Same strategy as above, but for a generic polynomial (no partially baked coefficients).
#
# Now fi[ i1_F_c ] is the constant term, fi[ i1_X ] is the coefficient of (x - xi), fi[ i2_X2 ] is the coefficient of (x - xi)**2, ...
#
cdef int general_1D( int order, double* fi, double xi, double[::view.generic] x, double* out ) nogil:
if order not in [0,1,2,3,4]:
return -1
# with gil:
# raise ValueError( "order must be 1, 2, 3 or 4; got %d" % order )
cdef int n = x.shape[0]
cdef int k
cdef double dx
cdef double acc # accumulator
if order == 4:
for k in range(n):
dx = x[k] - xi
acc = fma( dx, fi[ defs.i1_X4_c ], fi[ defs.i1_X3_c ] )
acc = fma( dx, acc, fi[ defs.i1_X2_c ] )
acc = fma( dx, acc, fi[ defs.i1_X_c ] )
out[k] = fma( dx, acc, fi[ defs.i1_F_c ] )
elif order == 3:
for k in range(n):
dx = x[k] - xi
acc = fma( dx, fi[ defs.i1_X3_c ], fi[ defs.i1_X2_c ] )
acc = fma( dx, acc, fi[ defs.i1_X_c ] )
out[k] = fma( dx, acc, fi[ defs.i1_F_c ] )
elif order == 2:
for k in range(n):
dx = x[k] - xi
acc = fma( dx, fi[ defs.i1_X2_c ], fi[ defs.i1_X_c ] )
out[k] = fma( dx, acc, fi[ defs.i1_F_c ] )
elif order == 1:
for k in range(n):
dx = x[k] - xi
out[k] = fma( dx, fi[ defs.i1_X_c ], fi[ defs.i1_F_c ] ) # fused multiply-add, op1*op2 + op3
else: # order == 0:
for k in range(n):
out[k] = fi[ defs.i1_F_c ]
return 0