A collection of fast numerical routines in C to compute functions associated with statistics on the sphere. Handmade Python interfaces that only require numpy are also provided.
- legendre_pl – Legendre polynomial as function of l
- wigner_3jj – Wigner 3j symbol as function of l1
- wigner_3jm – Wigner 3j symbol as function of m2
- wigner_6j – Wigner 6j symbol as function of l1
- wigner_dl – Wigner d function as function of l
void legendre_pl(int lmin, int lmax, double x, double* p) [source]
Compute the Legendre polynomials P_l(x) for all degrees l = lmin to l = lmax, with x being held fixed. The results are stored in the array p, which must have a size of at least lmax-lmin+1.
The code uses the well-known recurrence. This is the special case m1 = m2 = 0 of wigner_dl.
int wigner_3jj(double l2, double l3, double m2, double m3, double* l1min, double* l1max, double* thrcof, int ndim) [source]
Evaluate the Wigner 3j symbol
/ l1 l2 l3 \
\ -m2-m3 m2 m3 /
for all allowable values of l1, with the other parameters held fixed. For physically meaningful outputs, the arguments l2, l3, m2, m3 must be integer or half-integer, although other inputs are allowed. On return, l1min and l1max contain the smallest and largest allowable values of l1, respectively. The values of the 3j symbols are stored in the array pointed to by thrcof. Its allocated size is passed as ndim, which must be at least l1max-l1min+1. If thrcof is NULL, the function returns after checking the inputs and setting l1min and l1max, so that an array of appropriate size can be allocated. The function returns an error flag:
- 0 if no errors,
- 1 if either l2 < abs(m2) or l3 < abs(m3),
- 2 if either l2+abs(m2) or l3+abs(m3) non-integer,
- 3 if l1max-l1min not an integer,
- 4 if l1max less than l1min,
- 5 if ndim less than l1max-l1min+1.
The code is a reimplementation of the SLATEC routine drc3jj in C.
int wigner_3jm(double l1, double l2, double l3, double m1, double* m2min, double* m2max, double* thrcof, int ndim) [source]
Evaluate the Wigner 3j symbol
/ l1 l2 l3 \
\ m1 m2 -m1-m2 /
for all allowable values of m2, with the other parameters held fixed. For physically meaningful outputs, the arguments l1, l2, l3, m1 must be integer or half-integer, although other inputs are allowed. On return, m2min and m2max contain the smallest and largest allowable values of m2, respectively. The values of the 3j symbols are stored in the array pointed to by thrcof. Its allocated size is passed as ndim, which must be at least m2max-m2min+1. If thrcof is NULL, the function returns after checking the inputs and setting m2min and m2max, so that an array of appropriate size can be allocated. The function returns an error flag:
- 0 if no errors,
- 1 if either l1 < abs(m1) or l1+abs(m1) non-integer,
- 2 if abs(l1-l2) <= l3 <= l1+l2 not satisfied,
- 3 if l1+l2+l3 not an integer,
- 4 if m2max-m2min not an integer,
- 5 if m2max less than m2min,
- 6 if ndim less than m2max-m2min+1.
The code is a reimplementation of the SLATEC routine drc3jj in C.
int wigner_6j(double l2, double l3, double l4, double l5, double l6, double* l1min, double* l1max, double* sixcof, int ndim) [source]
Evaluate the Wigner 6j symbol
{l1 l2 l3}
{l4 l5 l6}
for all allowable values of l1, with the other parameters held fixed. For physically meaningful outputs, the arguments l2, l3, l4, l5, l6 must be integer or half-integer, although other inputs are allowed. On return, l1min and l1max contain the smallest and largest allowable values of l1, respectively. The values of the 6j symbols are stored in the array pointed to by sixcof. Its allocated size is passed as ndim, which must be at least l1max-l1min+1. If sixcof is NULL, the function returns after checking the inputs and setting l1min and l1max, so that an array of appropriate size can be allocated. The function returns an error flag:
- 0 if no errors,
- 1 if either l2+l3+l5+l6 or l4+l2+l6 not an integer,
- 2 if l4, l2, l6 triangular condition not satisfied,
- 3 if l4, l5, l3 triangular condition not satisfied,
- 4 if l1max-l1min not an integer,
- 5 if l1max less than l1min,
- 6 if ndim less than l1max-l1min+1.
The code is a reimplementation of the SLATEC routine drc6j in C.
void wigner_dl(int lmin, int lmax, int m1, int m2, double theta, double* d) [source]
Compute the Wigner d functions d^l_{m1, m2}(theta) for all degrees l = lmin to l = lmax, with m1, m2, and theta being held fixed. The angle theta is given in radian. The results are stored in the array d, which must have a size of at least lmax-lmin+1.
The code uses the recurrence described in arXiv:1904.09973. By default, SSE intrinsics are used to speed up the computation, if available, although this can be turned off at compile time using -DNOSSE.