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
<https://en.algorithmica.org/hpc/pipelining/branching/>.
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 / Cphim/u 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()
<https://github.com/geographiclib/geographiclib/blob/main/include/GeographicLib/Math.hpp#L280>
, Clenshaw()
<https://github.com/geographiclib/geographiclib/blob/main/src/AuxLatitude.cpp#L1319>).
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
<https://en.wikipedia.org/wiki/Horner%27s_method> or Clenshaw
<https://en.wikipedia.org/wiki/Clenshaw_algorithm>, 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.
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