Dear Even, Charles, Thomas, All,
Please find below a couple revised implementations of the authalic ==> geodetic
conversion using Horner's method and Clenshaw summation algorithm, both sharing
this table of coefficients from A20:
#define AUTH_ORDER 6
static const double Cphimu[21] = // Cφξ (A20) - coefficients to convert
authalic latitude to geodetic latitude
{
4 / 3.0, 4 / 45.0, -16/35.0, -2582 /14175.0, 60136 /467775.0,
28112932/ 212837625.0,
46 / 45.0, 152/945.0, -11966 /14175.0, -21016 / 51975.0,
251310128/ 638512875.0,
3044/2835.0, 3802 /14175.0, -94388 / 66825.0,
-8797648/ 10945935.0,
6059 / 4725.0, 41072 / 93555.0,
-1472637812/ 638512875.0,
768272 /467775.0,
-455935736/ 638512875.0,
4210684958/1915538625.0
};
This first one is using the existing functions from mlfn.cpp (untouched other
than possibly different formatting here):
// Evaluate sum(p[i] * x^i, i, 0, N) via Horner's method (p is of length N+1)
static inline double polyval(double x, const double p[], int N)
{
double y = N < 0 ? 0 : p[N];
while(N > 0)
y = y * x + p[--N];
return y;
}
// Evaluate y = sum(c[k] * sin((2*k+2) * zeta), k, 0, K-1)
static inline double clenshaw(double szeta, double czeta, const double c[], int
K)
{
// Approx operation count = (K + 5) mult and (2 * K + 2) add
double u0 = 0, u1 = 0; // accumulators for sum
double X = 2 * (czeta - szeta) * (czeta + szeta); // 2 * cos(2*zeta)
while(K > 0)
{
double t = X * u0 - u1 + c[--K];
u1 = u0;
u0 = t;
}
return 2 * szeta * czeta * u0; // sin(2*zeta) * u0
}
//https://arxiv.org/pdf/2212.05818
// ∆η(ζ) = S^(L)(ζ) · Cηζ · P^(M) (n) + O(n^L+1) -- (20)
void pj_authset(double a, double b, double cp[AUTH_ORDER])
{
double n = (a - b) / (a + b); // Third flattening
double d = n;
int l, o;
for(l = 0, o = 0; l < AUTH_ORDER; l++)
{
int m = AUTH_ORDER - l - 1;
cp[l] = d * polyval(n, Cphimu + o, m);
d *= n;
o += m + 1;
}
}
double pj_auth2geodlat(const double * cp, double phi)
{
return phi + clenshaw(sin(phi), cos(phi), cp, AUTH_ORDER);
}
For this second implementation, I unrolled the loops to get rid of the
iterations (and associated counter incrementations) and conditionals, which if
the compiler is not optimizing out, could potentially introduce some branching
costs.
This unrolled version remains quite compact (at least in this particular
formatting which the pre-commit hook will certainly massacre). The sequence of
operations is exactly the same, and I've tested that the two are equivalent,
and also equivalent with the two earlier implementations that I shared which
were not using Horner and Clenshaw, and also equivalent to 8 decimals to the
existing pj_authlat() function in PROJ.
//https://arxiv.org/pdf/2212.05818
// ∆η(ζ) = S^(L)(ζ) · Cηζ · P^(M) (n) + O(n^L+1) -- (20)
void pj_authset(double a, double b, double cp[AUTH_ORDER])
{
// Precomputing coefficient based on Horner's method
double n = (a - b) / (a + b); // Third flattening
const double * C = Cphimu;
double d = n;
cp[0] = (((((C[ 5] * n + C[ 4]) * n + C[ 3]) * n + C[ 2]) * n + C[ 1]) * n
+ C[ 0]) * d, d *= n;
cp[1] = (((( C[10] * n + C[ 9]) * n + C[ 8]) * n + C[ 7]) * n
+ C[ 6]) * d, d *= n;
cp[2] = ((( C[14] * n + C[13]) * n + C[12]) * n
+ C[11]) * d, d *= n;
cp[3] = (( C[17] * n + C[16]) * n
+ C[15]) * d, d *= n;
cp[4] = ( C[19] * n
+ C[18]) * d, d *= n;
cp[5] =
C[20] * d;
}
double pj_auth2geodlat(const double * cp, double phi)
{
// Using Clenshaw summation algorithm (order 6)
double szeta = sin(phi), czeta = cos(phi);
// Approx operation count = (K + 5) mult and (2 * K + 2) add
double X = 2 * (czeta - szeta) * (czeta + szeta); // 2 * cos(2*zeta)
double u0 = 0, u1 = 0; // accumulators for sum
double t;
t = X * u0 - u1 + cp[5], u1 = u0, u0 = t;
t = X * u0 - u1 + cp[4], u1 = u0, u0 = t;
t = X * u0 - u1 + cp[3], u1 = u0, u0 = t;
t = X * u0 - u1 + cp[2], u1 = u0, u0 = t;
t = X * u0 - u1 + cp[1], u1 = u0, u0 = t;
t = X * u0 - u1 + cp[0];
return phi + /* sin(2*zeta) * u0 */ 2 * szeta * czeta * t;
}
Note that the output of these two versions of pj_authset() (the 6 constants
precomputed from the authalic ==> geodetic A20 conversion matrix and the
ellipsoid's third flattening) is exactly the same as the previous version not
using Horner's method, and I believe also the same as the current output of
pj_autset() except that it currently uses only 3 constants for order 3 rather than
6 for order 6.
With both of these versions, we're down to only one sin() and one cos() call,
as per Even's suggestion, so I imagine that the Clenshaw algorithm does take
advantage of that trigonometric identity trick.
If we go with the separate polyval() and clenshaw() functions, then I suggest
we move these functions to a header file so that we can share them between
mlfn.cpp and auth.cpp while allowing the compiler to hopefully efficiently
inline them, and also hopefully optimize the code close to or equivalent to the
unrolled version (we could always compare the disassembly to verify whether
this is the case or not, but I would leave that to others).
My own preference would be for the unrolled version.
We could also make C / Cphimu a parameter to pj_authset() (which could be named
something else), since this could be used for other conversions between
auxiliary latitudes.
Similarly, pj_auth2geodlat() could actually be used for different conversions
if passing it pre-computed coefficients for other conversions, so perhaps it
could have a more generic names.
The rectifying latitude for pj_enfn() is a bit special because it uses n^2
rather than n, which tripped me up for a little while.
Thoughts / suggestions on how to move forward with this?
As a next step I would prepare a Pull Request based on your feedback, if you
have a preference for the shared functions or the unrolled loops approach.
Thank you very much for your help and guidance!
Kind regards,
-Jerome
On 9/11/24 4:21 PM, Jérôme St-Louis wrote:
So it seems like we already have an implementation of Horner and Clenshaw in:
https://github.com/OSGeo/PROJ/blob/master/src/mlfn.cpp
called polyval() and clenshaw() just like in GeographicLib ( polyval() ,
Clenshaw()).
It seems like Charles wrote or at least updated that :)
That is using the Cµφ (C[mu phi]) (A5) and Cφµ (C[phi mu]) (A6) from page 12 of the
paper, where µ is called the "rectifying latitude".
I imagine that this is directly related to the "meridional distance" ?
Perhaps we could re-organize this a bit to share this polyval() and clenshaw()
(they are currently static functions local to this mlfn.cpp) for use in
auth.cpp ?
Thanks!
Kind regards,
-Jerome
On 9/11/24 3:33 PM, Jérôme St-Louis wrote:
Thanks a lot for the input Charles and Thomas,
I am not familiar with either Horner or Clenshaw, but I do see the mentions now
on Section 6 - Evaluating the series pages 6 and 7 of the papers.
I implemented the simpler basic approach from section 3 / page 3, which also
happened to more easily correspond to the existing PROJ implementation.
I can definitely try to understand all this, with the help of this Rust Geodesy
code and the GeographicLib code, and have a go at updating my proposed
implementation for improved accuracy and performance.
Kind regards,
-Jerome
On 9/11/24 12:18 PM, Thomas Knudsen wrote:
I totally agree with Charles regarding using Horner for polynomial
evaluation and Clenshaw for the trig series - for accuracy and speed.
I implemented all the material from Charles' preprint
https://arxiv.org/pdf/2212.05818 for Rust Geodesy, when the preprint
appeared about 1½ years ago.
And although (being an experiment) my handling of the raw coefficients
is rather clumsy, at least it gave me a reason to revise my PROJ horner
and clenshaw implementations (which in turn were based on material from
Poder & Engsager: "Some Conformal Mappings...").
So Jérôme, perhaps take a look at the functions "taylor" and "fourier"
over athttps://github.com/busstoptaktik/geodesy/blob/main/src/math/series.rs
While written in Rust, translating to C++ should be rather trivial,
and they may be easier to follow than my decade-old versions already
in the PROJ code base.
