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qhull.pyx
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"""
Wrappers for Qhull triangulation, plus some additional N-D geometry utilities
.. versionadded:: 0.9
"""
#
# Copyright (C) Pauli Virtanen, 2010.
#
# Distributed under the same BSD license as Scipy.
#
import threading
import numpy as np
cimport numpy as np
cimport cython
cimport qhull
from numpy.compat import asbytes
__all__ = ['Delaunay', 'ConvexHull', 'Voronoi', 'tsearch']
#------------------------------------------------------------------------------
# Qhull interface
#------------------------------------------------------------------------------
cdef extern from "stdio.h":
extern void *stdin
extern void *stderr
extern void *stdout
cdef extern from "math.h":
double fabs(double x) nogil
double sqrt(double x) nogil
cdef extern from "setjmp.h" nogil:
ctypedef struct jmp_buf:
pass
int setjmp(jmp_buf STATE) nogil
void longjmp(jmp_buf STATE, int VALUE) nogil
cdef extern from "qhull/src/qset.h":
ctypedef union setelemT:
void *p
int i
ctypedef struct setT:
int maxsize
setelemT e[1]
int qh_setsize(setT *set) nogil
void qh_setappend(setT **setp, void *elem) nogil
cdef extern from "qhull/src/libqhull.h":
ctypedef double realT
ctypedef double coordT
ctypedef double pointT
ctypedef int boolT
ctypedef unsigned int flagT
ctypedef struct facetT:
coordT offset
coordT *center
coordT *normal
facetT *next
facetT *previous
unsigned id
setT *vertices
setT *neighbors
setT *ridges
setT *coplanarset
flagT simplicial
flagT flipped
flagT upperdelaunay
unsigned visitid
ctypedef struct vertexT:
vertexT *next
vertexT *previous
unsigned int id, visitid
pointT *point
setT *neighbors
ctypedef struct ridgeT:
setT *vertices
facetT *top
facetT *bottom
ctypedef struct qhT:
boolT DELAUNAY
boolT SCALElast
boolT KEEPcoplanar
boolT MERGEexact
boolT NOerrexit
boolT PROJECTdelaunay
boolT ATinfinity
boolT UPPERdelaunay
boolT hasTriangulation
int normal_size
char *qhull_command
facetT *facet_list
facetT *facet_tail
vertexT *vertex_list
vertexT *vertex_tail
int num_facets
int num_points
int center_size
unsigned int facet_id
pointT *first_point
pointT *input_points
realT last_low
realT last_high
realT last_newhigh
realT max_outside
realT MINoutside
realT DISTround
jmp_buf errexit
setT *other_points
extern qhT *qh_qh
extern int qh_PRINToff
extern int qh_ALL
void qh_init_A(void *inp, void *out, void *err, int argc, char **argv) nogil
void qh_init_B(realT *points, int numpoints, int dim, boolT ismalloc) nogil
void qh_checkflags(char *, char *) nogil
void qh_initflags(char *) nogil
void qh_option(char *, char*, char* ) nogil
void qh_freeqhull(boolT) nogil
void qh_memfreeshort(int *curlong, int *totlong) nogil
void qh_qhull() nogil
void qh_check_output() nogil
void qh_produce_output() nogil
void qh_triangulate() nogil
void qh_checkpolygon() nogil
void qh_findgood_all() nogil
void qh_appendprint(int format) nogil
setT *qh_pointvertex() nogil
realT *qh_readpoints(int* num, int *dim, boolT* ismalloc) nogil
int qh_new_qhull(int dim, int numpoints, realT *points,
boolT ismalloc, char* qhull_cmd, void *outfile,
void *errfile) nogil
int qh_pointid(pointT *point) nogil
vertexT *qh_nearvertex(facetT *facet, pointT *point, double *dist) nogil
boolT qh_addpoint(pointT *furthest, facetT *facet, boolT checkdist) nogil
facetT *qh_findbestfacet(pointT *point, boolT bestoutside,
realT *bestdist, boolT *isoutside) nogil
void qh_setdelaunay(int dim, int count, pointT *points) nogil
void qh_restore_qhull(qhT **oldqh) nogil
qhT *qh_save_qhull() nogil
cdef extern from "qhull/src/io.h":
ctypedef enum qh_RIDGE:
qh_RIDGEall
qh_RIDGEinner
qh_RIDGEouter
ctypedef void printvridgeT(void *fp, vertexT *vertex, vertexT *vertexA,
setT *centers, boolT unbounded)
int qh_eachvoronoi_all(void *fp, void* printvridge,
boolT isUpper, qh_RIDGE innerouter,
boolT inorder) nogil
void qh_order_vertexneighbors(vertexT *vertex) nogil
int qh_compare_facetvisit(void *p1, void *p2) nogil
cdef extern from "qhull/src/geom.h":
pointT *qh_facetcenter(setT *vertices) nogil
cdef extern from "qhull/src/poly.h":
void qh_check_maxout() nogil
cdef extern from "qhull/src/mem.h":
void qh_memfree(void *object, int insize)
from libc.string cimport memcpy
from libc.stdlib cimport qsort
#------------------------------------------------------------------------------
# LAPACK interface
#------------------------------------------------------------------------------
cdef extern from "qhull_blas.h":
void qh_dgetrf(int *m, int *n, double *a, int *lda, int *ipiv,
int *info) nogil
void qh_dgetrs(char *trans, int *n, int *nrhs, double *a, int *lda,
int *ipiv, double *b, int *ldb, int *info) nogil
void qh_dgecon(char *norm, int *n, double *a, int *lda, double *anorm,
double *rcond, double *work, int *iwork, int *info) nogil
#------------------------------------------------------------------------------
# Qhull wrapper
#------------------------------------------------------------------------------
# Qhull is not threadsafe: needs locking
_qhull_lock = threading.Lock()
# Qhull has (swappable) global state: keep track which Qhull instance is active
# and how many instances are alive
cdef _Qhull _active_qhull = None
cdef int _qhull_count = 0
class QhullError(RuntimeError):
pass
@cython.final
cdef class _Qhull:
cdef qhT *_saved_qh
cdef list _point_arrays
cdef public bytes options
cdef public bytes mode_option
cdef public object furthest_site
cdef int numpoints, ndim, _is_delaunay
cdef np.ndarray _ridge_points
cdef list _ridge_vertices
cdef object _ridge_error
cdef int _nridges
cdef np.ndarray _ridge_equations
@cython.final
def __init__(self,
bytes mode_option,
np.ndarray[np.double_t, ndim=2] points,
bytes options=None,
bytes required_options=None,
furthest_site=False,
incremental=False):
global _active_qhull, _qhull_count
cdef int exitcode
points = np.ascontiguousarray(points, dtype=np.double)
self.numpoints = points.shape[0]
self.ndim = points.shape[1]
if self.numpoints <= 0:
raise ValueError("No points given")
if self.ndim < 2:
raise ValueError("Need at least 2-D data")
# Process options
option_set = set()
if options is not None:
option_set.update(options.split())
if furthest_site:
if b"Qz" in option_set:
option_set.remove(b"Qz")
option_set.add(b"Qu")
if required_options is not None:
required_option_set = set(required_options.split())
if b"QJ" in option_set and b"Qt" in required_option_set:
# safe to remove, QJ always produces simplical output
required_option_set.remove(b"Qt")
option_set.update(required_option_set)
if incremental:
incremental_bad_ops = set([b'Qbb', b'Qbk', b'QBk', b'QbB', b'Qz'])
bad_opts = []
for bad_opt in incremental_bad_ops:
if bad_opt in options:
bad_opts.append(bad_opt)
if bad_opts:
raise ValueError("Qhull options %r are incompatible "
"with incremental mode" % (bad_opts,))
if b"Qt" in option_set:
# Qhull wants this
option_set.add(b"Q11")
# We need to own the copy of the points in incremental mode
points = points.copy()
if mode_option in (b"d", b"v"):
self._is_delaunay = 1
else:
self._is_delaunay = 0
self._point_arrays = [points]
self.options = b" ".join(option_set)
self.mode_option = mode_option
self.furthest_site = furthest_site
options = b"qhull " + mode_option + b" " + self.options
_qhull_lock.acquire()
try:
if _active_qhull is not None:
_active_qhull._deactivate()
_active_qhull = self
_qhull_count += 1
options_c = <char*>options
with nogil:
exitcode = qh_new_qhull(self.ndim, self.numpoints,
<realT*>points.data, 0,
options_c, NULL, stderr)
if exitcode != 0:
self._uninit()
raise QhullError("Qhull error")
finally:
_qhull_lock.release()
@cython.final
def close(self):
_qhull_lock.acquire()
try:
if _active_qhull is self or self._saved_qh != NULL:
self._uninit()
finally:
_qhull_lock.release()
@cython.final
def __del__(self):
self.close()
@cython.final
cdef int _activate(self) except -1:
"""
Activate this instance (_qhull_lock MUST be held when calling this)
"""
global _active_qhull
if _active_qhull is self:
return 0
elif _active_qhull is not None:
_active_qhull._deactivate()
assert _active_qhull is None
if self._saved_qh == NULL:
raise RuntimeError("Qhull instance is closed")
qh_restore_qhull(&self._saved_qh)
self._saved_qh = NULL
_active_qhull = self
return 0
@cython.final
cdef int _deactivate(self) except -1:
"""
Deactivate this instance (_qhull_lock MUST be held when calling this)
"""
global _active_qhull
assert _active_qhull is self
assert self._saved_qh == NULL
self._saved_qh = qh_save_qhull()
_active_qhull = None
@cython.final
cdef int _uninit(self) except -1:
"""
Uninitialize this instance (_qhull_lock MUST be held when calling this)
"""
global _active_qhull, _qhull_count
cdef int curlong, totlong
self._activate()
qh_freeqhull(qh_ALL)
_qhull_count -= 1
_active_qhull = None
self._saved_qh = NULL
if _qhull_count == 0:
# last one out cleans the house
qh_memfreeshort(&curlong, &totlong)
if curlong != 0 or totlong != 0:
raise QhullError(
"qhull: did not free %d bytes (%d pieces)" %
(totlong, curlong))
return 0
@cython.final
def get_points(self):
if len(self._point_arrays) == 1:
return self._point_arrays[0]
else:
return np.concatenate(
[x[:,:self.ndim] for x in self._point_arrays],
axis=0)
@cython.final
def add_points(self, points):
cdef int j
cdef realT *p
cdef facetT *facet
cdef double bestdist
cdef boolT isoutside
cdef np.ndarray arr
points = np.asarray(points)
if points.ndim!=2 or points.shape[1] != self._point_arrays[0].shape[1]:
raise ValueError("invalid size for new points array")
if points.size == 0:
return
if self._is_delaunay:
arr = np.empty((points.shape[0], self.ndim+1), dtype=np.double)
arr[:,:-1] = points
else:
arr = np.array(points, dtype=np.double, order="C", copy=True)
_qhull_lock.acquire()
try:
self._activate()
# nonlocal error handling
exitcode = setjmp(qh_qh.errexit)
if exitcode != 0:
raise QhullError("Qhull error")
qh_qh.NOerrexit = 0
# add points to triangulation
if self._is_delaunay:
# lift to paraboloid
qh_setdelaunay(arr.shape[1], arr.shape[0], <realT*>arr.data)
p = <realT*>arr.data
for j in xrange(arr.shape[0]):
facet = qh_findbestfacet(p, 0, &bestdist, &isoutside)
if isoutside:
if not qh_addpoint(p, facet, 0):
break
else:
# append the point to the "other points" list, to
# maintain the point IDs
qh_setappend(&qh_qh.other_points, p)
p += arr.shape[1]
qh_check_maxout()
qh_qh.hasTriangulation = 0
self._point_arrays.append(arr)
finally:
qh_qh.NOerrexit = 1
_qhull_lock.release()
@cython.final
def get_paraboloid_shift_scale(self):
cdef double paraboloid_scale
cdef double paraboloid_shift
_qhull_lock.acquire()
try:
self._activate()
if qh_qh.SCALElast:
paraboloid_scale = qh_qh.last_newhigh / (
qh_qh.last_high - qh_qh.last_low)
paraboloid_shift = - qh_qh.last_low * paraboloid_scale
else:
paraboloid_scale = 1.0
paraboloid_shift = 0.0
finally:
_qhull_lock.release()
return paraboloid_scale, paraboloid_shift
@cython.final
def triangulate(self):
_qhull_lock.acquire()
try:
self._activate()
with nogil:
qh_triangulate() # get rid of non-simplical facets
finally:
_qhull_lock.release()
@cython.final
def get_simplex_facet_array(self):
_qhull_lock.acquire()
try:
self._activate()
return self._get_simplex_facet_array()
finally:
_qhull_lock.release()
@cython.final
@cython.boundscheck(False)
@cython.cdivision(True)
cdef _get_simplex_facet_array(self):
"""
Return array of simplical facets currently in Qhull.
Returns
-------
facets : array of int, shape (nfacets, ndim+1)
Indices of coordinates of vertices forming the simplical facets
neighbors : array of int, shape (nfacets, ndim)
Indices of neighboring facets. The kth neighbor is opposite
the kth vertex, and the first neighbor is the horizon facet
for the first vertex.
Facets extending to infinity are denoted with index -1.
equations : array of double, shape (nfacets, ndim+2)
"""
cdef facetT* facet
cdef facetT* neighbor
cdef vertexT *vertex
cdef pointT *point
cdef int i, j, ipoint, ipoint2, ncoplanar
cdef object tmp
cdef np.ndarray[np.npy_int, ndim=2] facets
cdef np.ndarray[np.npy_int, ndim=2] neighbors
cdef np.ndarray[np.npy_int, ndim=2] coplanar
cdef np.ndarray[np.double_t, ndim=2] equations
cdef np.ndarray[np.npy_int, ndim=1] id_map
cdef double dist
cdef int facet_ndim
cdef int numpoints
facet_ndim = self.ndim
numpoints = self.numpoints
if self._is_delaunay:
facet_ndim += 1
id_map = np.empty((qh_qh.facet_id,), dtype=np.intc)
id_map.fill(-1)
# Compute facet indices
with nogil:
facet = qh_qh.facet_list
j = 0
while facet and facet.next:
if not self._is_delaunay or facet.upperdelaunay == qh_qh.UPPERdelaunay:
if not facet.simplicial and ( \
qh_setsize(facet.vertices) != facet_ndim or \
qh_setsize(facet.neighbors) != facet_ndim):
with gil:
raise QhullError(
"non-simplical facet encountered: %r vertices"
% (qh_setsize(facet.vertices),))
id_map[facet.id] = j
j += 1
facet = facet.next
# Allocate output
facets = np.zeros((j, facet_ndim), dtype=np.intc)
neighbors = np.zeros((j, facet_ndim), dtype=np.intc)
equations = np.zeros((j, facet_ndim+1), dtype=np.double)
ncoplanar = 0
coplanar = np.zeros((10, 3), dtype=np.intc)
# Retrieve facet information
with nogil:
facet = qh_qh.facet_list
j = 0
while facet and facet.next:
if self._is_delaunay and facet.upperdelaunay != qh_qh.UPPERdelaunay:
facet = facet.next
continue
# Save vertex info
for i in xrange(facet_ndim):
vertex = <vertexT*>facet.vertices.e[i].p
ipoint = qh_pointid(vertex.point)
facets[j, i] = ipoint
# Save neighbor info
for i in xrange(facet_ndim):
neighbor = <facetT*>facet.neighbors.e[i].p
neighbors[j,i] = id_map[neighbor.id]
# Save simplex equation info
for i in xrange(facet_ndim):
equations[j,i] = facet.normal[i]
equations[j,facet_ndim] = facet.offset
# Save coplanar info
if facet.coplanarset:
for i in range(qh_setsize(facet.coplanarset)):
point = <pointT*>facet.coplanarset.e[i].p
vertex = qh_nearvertex(facet, point, &dist)
if ncoplanar >= coplanar.shape[0]:
with gil:
tmp = coplanar
coplanar = None
try:
tmp.resize(2*ncoplanar+1, 3)
except ValueError:
# Work around Cython issue on Python 2.4
tmp = np.resize(tmp, (2*ncoplanar+1, 3))
coplanar = tmp
coplanar[ncoplanar,0] = qh_pointid(point)
coplanar[ncoplanar,1] = id_map[facet.id]
coplanar[ncoplanar,2] = qh_pointid(vertex.point)
ncoplanar += 1
j += 1
facet = facet.next
return facets, neighbors, equations, coplanar[:ncoplanar]
@cython.final
def get_voronoi_diagram(_Qhull self):
_qhull_lock.acquire()
try:
self._activate()
return self._get_voronoi_diagram()
finally:
_qhull_lock.release()
@cython.final
@cython.boundscheck(False)
@cython.cdivision(True)
cdef _get_voronoi_diagram(_Qhull self):
"""
Return the voronoi diagram currently in Qhull.
Returns
-------
voronoi_vertices : array of double, shape (nvoronoi_vertices, ndim)
Coordinates of the Voronoi vertices
ridge_points : array of double, shape (nridges, 2)
Voronoi ridges, as indices to the points array.
ridge_vertices : list of lists, shape (nridges, *)
Voronoi vertices for each Voronoi ridge, as indices to
the Voronoi vertices array.
Infinity is indicated by index ``-1``.
regions : list of lists, shape (nregion, *)
Voronoi vertices of all regions.
point_region : array of int, shape (npoint,)
Index of the Voronoi region for each input point.
"""
cdef int i, j, k
cdef vertexT *vertex
cdef facetT *neighbor
cdef facetT *facet
cdef object tmp
cdef np.ndarray[np.double_t, ndim=2] voronoi_vertices
cdef np.ndarray[np.intp_t, ndim=1] point_region
cdef int nvoronoi_vertices
cdef pointT infty_point[NPY_MAXDIMS+1]
cdef pointT *point
cdef pointT *center
cdef double dist
cdef int inf_seen
cdef list regions
cdef list cur_region
# -- Grab Voronoi ridges
self._nridges = 0
self._ridge_error = None
self._ridge_points = np.empty((10, 2), np.intc)
self._ridge_vertices = []
qh_eachvoronoi_all(<void*>self, &_visit_voronoi, qh_qh.UPPERdelaunay,
qh_RIDGEall, 1)
self._ridge_points = self._ridge_points[:self._nridges]
if self._ridge_error is not None:
raise self._ridge_error
# Now, qh_eachvoronoi_all has initialized the visitids of facets
# to correspond do the Voronoi vertex indices.
# -- Grab Voronoi regions
regions = []
point_region = np.empty((self.numpoints,), np.intp)
point_region.fill(-1)
vertex = qh_qh.vertex_list
while vertex and vertex.next:
qh_order_vertexneighbors_nd(self.ndim+1, vertex)
i = qh_pointid(vertex.point)
if i < self.numpoints:
# Qz results to one extra point
point_region[i] = len(regions)
inf_seen = 0
cur_region = []
for k in xrange(qh_setsize(vertex.neighbors)):
neighbor = <facetT*>vertex.neighbors.e[k].p
i = neighbor.visitid - 1
if i == -1:
if not inf_seen:
inf_seen = 1
else:
continue
cur_region.append(int(i))
if len(cur_region) == 1 and cur_region[0] == -1:
# report similarly as qvoronoi o
cur_region = []
regions.append(cur_region)
vertex = vertex.next
# -- Grab Voronoi vertices and point-to-region map
nvoronoi_vertices = 0
voronoi_vertices = np.empty((10, self.ndim), np.double)
facet = qh_qh.facet_list
while facet and facet.next:
if facet.visitid > 0:
# finite Voronoi vertex
center = facet.center
if center == NULL:
center = qh_facetcenter(facet.vertices)
nvoronoi_vertices = max(facet.visitid, nvoronoi_vertices)
if nvoronoi_vertices >= voronoi_vertices.shape[0]:
tmp = voronoi_vertices
voronoi_vertices = None
try:
tmp.resize(2*nvoronoi_vertices + 1, self.ndim)
except ValueError:
tmp = np.resize(tmp, (2*nvoronoi_vertices+1, self.ndim))
voronoi_vertices = tmp
for k in range(self.ndim):
voronoi_vertices[facet.visitid-1, k] = center[k]
if center != facet.center:
qh_memfree(center, qh_qh.center_size)
if facet.coplanarset:
for k in range(qh_setsize(facet.coplanarset)):
point = <pointT*>facet.coplanarset.e[k].p
vertex = qh_nearvertex(facet, point, &dist)
i = qh_pointid(point)
j = qh_pointid(vertex.point)
if i < self.numpoints:
# Qz can result to one extra point
point_region[i] = point_region[j]
facet = facet.next
voronoi_vertices = voronoi_vertices[:nvoronoi_vertices]
return voronoi_vertices, self._ridge_points, self._ridge_vertices, \
regions, point_region
cdef void _visit_voronoi(void *ptr, vertexT *vertex, vertexT *vertexA,
setT *centers, boolT unbounded):
cdef _Qhull qh = <_Qhull>ptr
cdef int point_1, point_2, ix
cdef list cur_vertices
if qh._ridge_error is not None:
return
if qh._nridges >= qh._ridge_points.shape[0]:
try:
qh._ridge_points.resize(2*qh._nridges + 1, 2)
except Exception, e:
qh._ridge_error = e
return
# Record which points the ridge is between
point_1 = qh_pointid(vertex.point)
point_2 = qh_pointid(vertexA.point)
p = <int*>qh._ridge_points.data
p[2*qh._nridges + 0] = point_1
p[2*qh._nridges + 1] = point_2
# Record which voronoi vertices constitute the ridge
cur_vertices = []
for i in xrange(qh_setsize(centers)):
ix = (<facetT*>centers.e[i].p).visitid - 1
cur_vertices.append(ix)
qh._ridge_vertices.append(cur_vertices)
qh._nridges += 1
return
cdef void qh_order_vertexneighbors_nd(int nd, vertexT *vertex):
if nd == 3:
qh_order_vertexneighbors(vertex)
elif nd >= 4:
qsort(<facetT**>&vertex.neighbors.e[0].p, qh_setsize(vertex.neighbors),
sizeof(facetT*), qh_compare_facetvisit)
#------------------------------------------------------------------------------
# Barycentric coordinates
#------------------------------------------------------------------------------
@cython.boundscheck(False)
@cython.cdivision(True)
def _get_barycentric_transforms(np.ndarray[np.double_t, ndim=2] points,
np.ndarray[np.npy_int, ndim=2] simplices,
double eps):
"""
Compute barycentric affine coordinate transformations for given
simplices.
Returns
-------
Tinvs : array, shape (nsimplex, ndim+1, ndim)
Barycentric transforms for each simplex.
Tinvs[i,:ndim,:ndim] contains inverse of the matrix ``T``,
and Tinvs[i,ndim,:] contains the vector ``r_n`` (see below).
Notes
-----
Barycentric transform from ``x`` to ``c`` is defined by::
T c = x - r_n
where the ``r_1, ..., r_n`` are the vertices of the simplex.
The matrix ``T`` is defined by the condition::
T e_j = r_j - r_n
where ``e_j`` is the unit axis vector, e.g, ``e_2 = [0,1,0,0,...]``
This implies that ``T_ij = (r_j - r_n)_i``.
For the barycentric transforms, we need to compute the inverse
matrix ``T^-1`` and store the vectors ``r_n`` for each vertex.
These are stacked into the `Tinvs` returned.
"""
cdef np.ndarray[np.double_t, ndim=2] T
cdef np.ndarray[np.double_t, ndim=3] Tinvs
cdef int isimplex
cdef int i, j, n, nrhs, lda, ldb, info
cdef int ipiv[NPY_MAXDIMS+1]
cdef int ndim, nsimplex
cdef double centroid[NPY_MAXDIMS], c[NPY_MAXDIMS+1]
cdef double *transform
cdef double anorm, rcond
cdef double nan, rcond_limit
cdef double work[4*NPY_MAXDIMS]
cdef int iwork[NPY_MAXDIMS]
cdef double x1, x2, x3
cdef double y1, y2, y3
cdef double det
nan = np.nan
ndim = points.shape[1]
nsimplex = simplices.shape[0]
T = np.zeros((ndim, ndim), dtype=np.double)
Tinvs = np.zeros((nsimplex, ndim+1, ndim), dtype=np.double)
# Maximum inverse condition number to allow: we want at least three
# of the digits be significant, to be safe
rcond_limit = 1000*eps
with nogil:
for isimplex in xrange(nsimplex):
for i in xrange(ndim):
Tinvs[isimplex,ndim,i] = points[simplices[isimplex,ndim],i]
for j in xrange(ndim):
T[i,j] = (points[simplices[isimplex,j],i]
- Tinvs[isimplex,ndim,i])
Tinvs[isimplex,i,i] = 1
# compute 1-norm for estimating condition number
anorm = _matrix_norm1(ndim, <double*>T.data)
# LU decomposition
n = ndim
nrhs = ndim
lda = ndim
ldb = ndim
qh_dgetrf(&n, &n, <double*>T.data, &lda, ipiv, &info)
# Check condition number
if info == 0:
qh_dgecon("1", &n, <double*>T.data, &lda, &anorm, &rcond,
work, iwork, &info)
if rcond < rcond_limit:
# The transform seems singular
info = 1
# Compute transform
if info == 0:
qh_dgetrs("N", &n, &nrhs, <double*>T.data, &lda, ipiv,
(<double*>Tinvs.data) + ndim*(ndim+1)*isimplex,
&ldb, &info)
# Deal with degenerate simplices
if info != 0:
for i in range(ndim+1):
for j in range(ndim):
Tinvs[isimplex,i,j] = nan
return Tinvs
@cython.boundscheck(False)
cdef double _matrix_norm1(int n, double *a) nogil:
"""Compute the 1-norm of a square matrix given in in Fortran order"""
cdef double maxsum = 0, colsum
cdef int i, j
for j in range(n):
colsum = 0
for i in range(n):
colsum += fabs(a[0])
a += 1
if maxsum < colsum:
maxsum = colsum
return maxsum
cdef int _barycentric_inside(int ndim, double *transform,
double *x, double *c, double eps) nogil:
"""
Check whether point is inside a simplex, using barycentric
coordinates. `c` will be filled with barycentric coordinates, if
the point happens to be inside.
"""
cdef int i, j
c[ndim] = 1.0
for i in xrange(ndim):
c[i] = 0
for j in xrange(ndim):
c[i] += transform[ndim*i + j] * (x[j] - transform[ndim*ndim + j])
c[ndim] -= c[i]
if not (-eps <= c[i] <= 1 + eps):
return 0
if not (-eps <= c[ndim] <= 1 + eps):
return 0
return 1
cdef void _barycentric_coordinate_single(int ndim, double *transform,
double *x, double *c, int i) nogil:
"""
Compute a single barycentric coordinate.
Before the ndim+1'th coordinate can be computed, the other must have
been computed earlier.
"""
cdef int j
if i == ndim:
c[ndim] = 1.0
for j in xrange(ndim):
c[ndim] -= c[j]
else:
c[i] = 0
for j in xrange(ndim):
c[i] += transform[ndim*i + j] * (x[j] - transform[ndim*ndim + j])
cdef void _barycentric_coordinates(int ndim, double *transform,
double *x, double *c) nogil:
"""
Compute barycentric coordinates.
"""
cdef int i, j
c[ndim] = 1.0
for i in xrange(ndim):
c[i] = 0
for j in xrange(ndim):
c[i] += transform[ndim*i + j] * (x[j] - transform[ndim*ndim + j])
c[ndim] -= c[i]
#------------------------------------------------------------------------------
# N-D geometry
#------------------------------------------------------------------------------
cdef void _lift_point(DelaunayInfo_t *d, double *x, double *z) nogil:
cdef int i
z[d.ndim] = 0
for i in xrange(d.ndim):