511 lines
20 KiB
C++
511 lines
20 KiB
C++
/**
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* \file SphericalEngine.cpp
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* \brief Implementation for GeographicLib::SphericalEngine class
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*
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* Copyright (c) Charles Karney (2011-2022) <charles@karney.com> and licensed
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* under the MIT/X11 License. For more information, see
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* https://geographiclib.sourceforge.io/
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*
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* The general sum is\verbatim
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V(r, theta, lambda) = sum(n = 0..N) sum(m = 0..n)
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q^(n+1) * (C[n,m] * cos(m*lambda) + S[n,m] * sin(m*lambda)) * P[n,m](t)
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\endverbatim
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* where <tt>t = cos(theta)</tt>, <tt>q = a/r</tt>. In addition write <tt>u =
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* sin(theta)</tt>.
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*
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* <tt>P[n,m]</tt> is a normalized associated Legendre function of degree
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* <tt>n</tt> and order <tt>m</tt>. Here the formulas are given for full
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* normalized functions (usually denoted <tt>Pbar</tt>).
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*
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* Rewrite outer sum\verbatim
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V(r, theta, lambda) = sum(m = 0..N) * P[m,m](t) * q^(m+1) *
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[Sc[m] * cos(m*lambda) + Ss[m] * sin(m*lambda)]
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\endverbatim
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* where the inner sums are\verbatim
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Sc[m] = sum(n = m..N) q^(n-m) * C[n,m] * P[n,m](t)/P[m,m](t)
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Ss[m] = sum(n = m..N) q^(n-m) * S[n,m] * P[n,m](t)/P[m,m](t)
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\endverbatim
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* Evaluate sums via Clenshaw method. The overall framework is similar to
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* Deakin with the following changes:
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* - Clenshaw summation is used to roll the computation of
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* <tt>cos(m*lambda)</tt> and <tt>sin(m*lambda)</tt> into the evaluation of
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* the outer sum (rather than independently computing an array of these
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* trigonometric terms).
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* - Scale the coefficients to guard against overflow when <tt>N</tt> is large.
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* .
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* For the general framework of Clenshaw, see
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* http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
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*
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* Let\verbatim
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S = sum(k = 0..N) c[k] * F[k](x)
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F[n+1](x) = alpha[n](x) * F[n](x) + beta[n](x) * F[n-1](x)
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\endverbatim
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* Evaluate <tt>S</tt> with\verbatim
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y[N+2] = y[N+1] = 0
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y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]
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S = c[0] * F[0] + y[1] * F[1] + beta[1] * F[0] * y[2]
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\endverbatim
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* \e IF <tt>F[0](x) = 1</tt> and <tt>beta(0,x) = 0</tt>, then <tt>F[1](x) =
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* alpha(0,x)</tt> and we can continue the recursion for <tt>y[k]</tt> until
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* <tt>y[0]</tt>, giving\verbatim
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S = y[0]
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\endverbatim
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*
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* Evaluating the inner sum\verbatim
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l = n-m; n = l+m
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Sc[m] = sum(l = 0..N-m) C[l+m,m] * q^l * P[l+m,m](t)/P[m,m](t)
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F[l] = q^l * P[l+m,m](t)/P[m,m](t)
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\endverbatim
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* Holmes + Featherstone, Eq. (11), give\verbatim
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P[n,m] = sqrt((2*n-1)*(2*n+1)/((n-m)*(n+m))) * t * P[n-1,m] -
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sqrt((2*n+1)*(n+m-1)*(n-m-1)/((n-m)*(n+m)*(2*n-3))) * P[n-2,m]
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\endverbatim
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* thus\verbatim
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alpha[l] = t * q * sqrt(((2*n+1)*(2*n+3))/
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((n-m+1)*(n+m+1)))
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beta[l+1] = - q^2 * sqrt(((n-m+1)*(n+m+1)*(2*n+5))/
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((n-m+2)*(n+m+2)*(2*n+1)))
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\endverbatim
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* In this case, <tt>F[0] = 1</tt> and <tt>beta[0] = 0</tt>, so the <tt>Sc[m]
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* = y[0]</tt>.
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*
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* Evaluating the outer sum\verbatim
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V = sum(m = 0..N) Sc[m] * q^(m+1) * cos(m*lambda) * P[m,m](t)
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+ sum(m = 0..N) Ss[m] * q^(m+1) * cos(m*lambda) * P[m,m](t)
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F[m] = q^(m+1) * cos(m*lambda) * P[m,m](t) [or sin(m*lambda)]
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\endverbatim
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* Holmes + Featherstone, Eq. (13), give\verbatim
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P[m,m] = u * sqrt((2*m+1)/((m>1?2:1)*m)) * P[m-1,m-1]
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\endverbatim
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* also, we have\verbatim
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cos((m+1)*lambda) = 2*cos(lambda)*cos(m*lambda) - cos((m-1)*lambda)
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\endverbatim
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* thus\verbatim
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alpha[m] = 2*cos(lambda) * sqrt((2*m+3)/(2*(m+1))) * u * q
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= cos(lambda) * sqrt( 2*(2*m+3)/(m+1) ) * u * q
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beta[m+1] = -sqrt((2*m+3)*(2*m+5)/(4*(m+1)*(m+2))) * u^2 * q^2
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* (m == 0 ? sqrt(2) : 1)
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\endverbatim
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* Thus\verbatim
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F[0] = q [or 0]
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F[1] = cos(lambda) * sqrt(3) * u * q^2 [or sin(lambda)]
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beta[1] = - sqrt(15/4) * u^2 * q^2
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\endverbatim
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*
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* Here is how the various components of the gradient are computed
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*
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* Differentiate wrt <tt>r</tt>\verbatim
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d q^(n+1) / dr = (-1/r) * (n+1) * q^(n+1)
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\endverbatim
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* so multiply <tt>C[n,m]</tt> by <tt>n+1</tt> in inner sum and multiply the
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* sum by <tt>-1/r</tt>.
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*
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* Differentiate wrt <tt>lambda</tt>\verbatim
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d cos(m*lambda) = -m * sin(m*lambda)
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d sin(m*lambda) = m * cos(m*lambda)
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\endverbatim
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* so multiply terms by <tt>m</tt> in outer sum and swap sine and cosine
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* variables.
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*
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* Differentiate wrt <tt>theta</tt>\verbatim
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dV/dtheta = V' = -u * dV/dt = -u * V'
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\endverbatim
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* here <tt>'</tt> denotes differentiation wrt <tt>theta</tt>.\verbatim
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d/dtheta (Sc[m] * P[m,m](t)) = Sc'[m] * P[m,m](t) + Sc[m] * P'[m,m](t)
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\endverbatim
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* Now <tt>P[m,m](t) = const * u^m</tt>, so <tt>P'[m,m](t) = m * t/u *
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* P[m,m](t)</tt>, thus\verbatim
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d/dtheta (Sc[m] * P[m,m](t)) = (Sc'[m] + m * t/u * Sc[m]) * P[m,m](t)
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\endverbatim
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* Clenshaw recursion for <tt>Sc[m]</tt> reads\verbatim
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y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]
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\endverbatim
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* Substituting <tt>alpha[k] = const * t</tt>, <tt>alpha'[k] = -u/t *
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* alpha[k]</tt>, <tt>beta'[k] = c'[k] = 0</tt> gives\verbatim
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y'[k] = alpha[k] * y'[k+1] + beta[k+1] * y'[k+2] - u/t * alpha[k] * y[k+1]
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\endverbatim
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*
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* Finally, given the derivatives of <tt>V</tt>, we can compute the components
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* of the gradient in spherical coordinates and transform the result into
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* cartesian coordinates.
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**********************************************************************/
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#include <GeographicLib/SphericalEngine.hpp>
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#include <GeographicLib/CircularEngine.hpp>
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#include <GeographicLib/Utility.hpp>
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#if defined(_MSC_VER)
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// Squelch warnings about constant conditional expressions and potentially
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// uninitialized local variables
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# pragma warning (disable: 4127 4701)
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#endif
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namespace GeographicLib {
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using namespace std;
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vector<Math::real>& SphericalEngine::sqrttable() {
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static vector<real> sqrttable(0);
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return sqrttable;
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}
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template<bool gradp, SphericalEngine::normalization norm, int L>
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Math::real SphericalEngine::Value(const coeff c[], const real f[],
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real x, real y, real z, real a,
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real& gradx, real& grady, real& gradz)
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{
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static_assert(L > 0, "L must be positive");
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static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
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int N = c[0].nmx(), M = c[0].mmx();
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real
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p = hypot(x, y),
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cl = p != 0 ? x / p : 1, // cos(lambda); at pole, pick lambda = 0
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sl = p != 0 ? y / p : 0, // sin(lambda)
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r = hypot(z, p),
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t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
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u = r != 0 ? fmax(p / r, eps()) : 1, // sin(theta); but avoid the pole
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q = a / r;
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real
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q2 = Math::sq(q),
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uq = u * q,
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uq2 = Math::sq(uq),
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tu = t / u;
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// Initialize outer sum
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real vc = 0, vc2 = 0, vs = 0, vs2 = 0; // v [N + 1], v [N + 2]
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// vr, vt, vl and similar w variable accumulate the sums for the
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// derivatives wrt r, theta, and lambda, respectively.
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real vrc = 0, vrc2 = 0, vrs = 0, vrs2 = 0; // vr[N + 1], vr[N + 2]
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real vtc = 0, vtc2 = 0, vts = 0, vts2 = 0; // vt[N + 1], vt[N + 2]
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real vlc = 0, vlc2 = 0, vls = 0, vls2 = 0; // vl[N + 1], vl[N + 2]
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int k[L];
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const vector<real>& root( sqrttable() );
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for (int m = M; m >= 0; --m) { // m = M .. 0
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// Initialize inner sum
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real
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wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
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wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
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wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
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for (int l = 0; l < L; ++l)
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k[l] = c[l].index(N, m) + 1;
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for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
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real w, A, Ax, B, R; // alpha[l], beta[l + 1]
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switch (norm) {
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case FULL:
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w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
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Ax = q * w * root[2 * n + 3];
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A = t * Ax;
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B = - q2 * root[2 * n + 5] /
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(w * root[n - m + 2] * root[n + m + 2]);
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break;
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case SCHMIDT:
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w = root[n - m + 1] * root[n + m + 1];
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Ax = q * (2 * n + 1) / w;
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A = t * Ax;
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B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
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break;
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default: break; // To suppress warning message from Visual Studio
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}
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R = c[0].Cv(--k[0]);
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for (int l = 1; l < L; ++l)
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R += c[l].Cv(--k[l], n, m, f[l]);
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R *= scale();
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w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
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if (gradp) {
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w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
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w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
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}
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if (m) {
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R = c[0].Sv(k[0]);
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for (int l = 1; l < L; ++l)
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R += c[l].Sv(k[l], n, m, f[l]);
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R *= scale();
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w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
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if (gradp) {
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w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
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w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
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}
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}
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}
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// Now Sc[m] = wc, Ss[m] = ws
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// Sc'[m] = wtc, Ss'[m] = wtc
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if (m) {
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real v, A, B; // alpha[m], beta[m + 1]
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switch (norm) {
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case FULL:
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v = root[2] * root[2 * m + 3] / root[m + 1];
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A = cl * v * uq;
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B = - v * root[2 * m + 5] / (root[8] * root[m + 2]) * uq2;
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break;
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case SCHMIDT:
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v = root[2] * root[2 * m + 1] / root[m + 1];
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A = cl * v * uq;
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B = - v * root[2 * m + 3] / (root[8] * root[m + 2]) * uq2;
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break;
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default: break; // To suppress warning message from Visual Studio
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}
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v = A * vc + B * vc2 + wc ; vc2 = vc ; vc = v;
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v = A * vs + B * vs2 + ws ; vs2 = vs ; vs = v;
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if (gradp) {
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// Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
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wtc += m * tu * wc; wts += m * tu * ws;
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v = A * vrc + B * vrc2 + wrc; vrc2 = vrc; vrc = v;
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v = A * vrs + B * vrs2 + wrs; vrs2 = vrs; vrs = v;
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v = A * vtc + B * vtc2 + wtc; vtc2 = vtc; vtc = v;
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v = A * vts + B * vts2 + wts; vts2 = vts; vts = v;
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v = A * vlc + B * vlc2 + m*ws; vlc2 = vlc; vlc = v;
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v = A * vls + B * vls2 - m*wc; vls2 = vls; vls = v;
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}
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} else {
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real A, B, qs;
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switch (norm) {
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case FULL:
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A = root[3] * uq; // F[1]/(q*cl) or F[1]/(q*sl)
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B = - root[15]/2 * uq2; // beta[1]/q
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break;
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case SCHMIDT:
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A = uq;
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B = - root[3]/2 * uq2;
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break;
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default: break; // To suppress warning message from Visual Studio
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}
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qs = q / scale();
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vc = qs * (wc + A * (cl * vc + sl * vs ) + B * vc2);
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if (gradp) {
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qs /= r;
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// The components of the gradient in spherical coordinates are
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// r: dV/dr
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// theta: 1/r * dV/dtheta
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// lambda: 1/(r*u) * dV/dlambda
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vrc = - qs * (wrc + A * (cl * vrc + sl * vrs) + B * vrc2);
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vtc = qs * (wtc + A * (cl * vtc + sl * vts) + B * vtc2);
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vlc = qs / u * ( A * (cl * vlc + sl * vls) + B * vlc2);
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}
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}
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}
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if (gradp) {
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// Rotate into cartesian (geocentric) coordinates
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gradx = cl * (u * vrc + t * vtc) - sl * vlc;
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grady = sl * (u * vrc + t * vtc) + cl * vlc;
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gradz = t * vrc - u * vtc ;
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}
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return vc;
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}
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template<bool gradp, SphericalEngine::normalization norm, int L>
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CircularEngine SphericalEngine::Circle(const coeff c[], const real f[],
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real p, real z, real a) {
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static_assert(L > 0, "L must be positive");
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static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
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int N = c[0].nmx(), M = c[0].mmx();
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real
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r = hypot(z, p),
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t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
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u = r != 0 ? fmax(p / r, eps()) : 1, // sin(theta); but avoid the pole
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q = a / r;
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real
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q2 = Math::sq(q),
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tu = t / u;
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CircularEngine circ(M, gradp, norm, a, r, u, t);
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int k[L];
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const vector<real>& root( sqrttable() );
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for (int m = M; m >= 0; --m) { // m = M .. 0
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// Initialize inner sum
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real
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wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
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wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
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wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
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for (int l = 0; l < L; ++l)
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k[l] = c[l].index(N, m) + 1;
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for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
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real w, A, Ax, B, R; // alpha[l], beta[l + 1]
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switch (norm) {
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case FULL:
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w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
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Ax = q * w * root[2 * n + 3];
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A = t * Ax;
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B = - q2 * root[2 * n + 5] /
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(w * root[n - m + 2] * root[n + m + 2]);
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break;
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case SCHMIDT:
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w = root[n - m + 1] * root[n + m + 1];
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Ax = q * (2 * n + 1) / w;
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A = t * Ax;
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B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
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break;
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default: break; // To suppress warning message from Visual Studio
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}
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R = c[0].Cv(--k[0]);
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for (int l = 1; l < L; ++l)
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R += c[l].Cv(--k[l], n, m, f[l]);
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R *= scale();
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w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
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if (gradp) {
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w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
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w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
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}
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if (m) {
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R = c[0].Sv(k[0]);
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for (int l = 1; l < L; ++l)
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R += c[l].Sv(k[l], n, m, f[l]);
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R *= scale();
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w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
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if (gradp) {
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w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
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w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
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}
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}
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}
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if (!gradp)
|
|
circ.SetCoeff(m, wc, ws);
|
|
else {
|
|
// Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
|
|
wtc += m * tu * wc; wts += m * tu * ws;
|
|
circ.SetCoeff(m, wc, ws, wrc, wrs, wtc, wts);
|
|
}
|
|
}
|
|
|
|
return circ;
|
|
}
|
|
|
|
void SphericalEngine::RootTable(int N) {
|
|
// Need square roots up to max(2 * N + 5, 15).
|
|
vector<real>& root( sqrttable() );
|
|
int L = max(2 * N + 5, 15) + 1, oldL = int(root.size());
|
|
if (oldL >= L)
|
|
return;
|
|
root.resize(L);
|
|
for (int l = oldL; l < L; ++l)
|
|
root[l] = sqrt(real(l));
|
|
}
|
|
|
|
void SphericalEngine::coeff::readcoeffs(istream& stream, int& N, int& M,
|
|
vector<real>& C,
|
|
vector<real>& S,
|
|
bool truncate) {
|
|
if (truncate) {
|
|
if (!((N >= M && M >= 0) || (N == -1 && M == -1)))
|
|
// The last condition is that M = -1 implies N = -1.
|
|
throw GeographicErr("Bad requested degree and order " +
|
|
Utility::str(N) + " " + Utility::str(M));
|
|
}
|
|
int nm[2];
|
|
Utility::readarray<int, int, false>(stream, nm, 2);
|
|
int N0 = nm[0], M0 = nm[1];
|
|
if (!((N0 >= M0 && M0 >= 0) || (N0 == -1 && M0 == -1)))
|
|
// The last condition is that M0 = -1 implies N0 = -1.
|
|
throw GeographicErr("Bad degree and order " +
|
|
Utility::str(N0) + " " + Utility::str(M0));
|
|
N = truncate ? min(N, N0) : N0;
|
|
M = truncate ? min(M, M0) : M0;
|
|
C.resize(SphericalEngine::coeff::Csize(N, M));
|
|
S.resize(SphericalEngine::coeff::Ssize(N, M));
|
|
int skip = (SphericalEngine::coeff::Csize(N0, M0) -
|
|
SphericalEngine::coeff::Csize(N0, M )) * sizeof(double);
|
|
if (N == N0) {
|
|
Utility::readarray<double, real, false>(stream, C);
|
|
if (skip) stream.seekg(streamoff(skip), ios::cur);
|
|
Utility::readarray<double, real, false>(stream, S);
|
|
if (skip) stream.seekg(streamoff(skip), ios::cur);
|
|
} else {
|
|
for (int m = 0, k = 0; m <= M; ++m) {
|
|
Utility::readarray<double, real, false>(stream, &C[k], N + 1 - m);
|
|
stream.seekg((N0 - N) * sizeof(double), ios::cur);
|
|
k += N + 1 - m;
|
|
}
|
|
if (skip) stream.seekg(streamoff(skip), ios::cur);
|
|
for (int m = 1, k = 0; m <= M; ++m) {
|
|
Utility::readarray<double, real, false>(stream, &S[k], N + 1 - m);
|
|
stream.seekg((N0 - N) * sizeof(double), ios::cur);
|
|
k += N + 1 - m;
|
|
}
|
|
if (skip) stream.seekg(streamoff(skip), ios::cur);
|
|
}
|
|
return;
|
|
}
|
|
|
|
/// \cond SKIP
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<true, SphericalEngine::FULL, 1>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<false, SphericalEngine::FULL, 1>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 1>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 1>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<true, SphericalEngine::FULL, 2>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<false, SphericalEngine::FULL, 2>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 2>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 2>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<true, SphericalEngine::FULL, 3>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<false, SphericalEngine::FULL, 3>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 3>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
template Math::real GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 3>
|
|
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
|
|
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<true, SphericalEngine::FULL, 1>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<false, SphericalEngine::FULL, 1>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 1>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 1>
|
|
(const coeff[], const real[], real, real, real);
|
|
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<true, SphericalEngine::FULL, 2>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<false, SphericalEngine::FULL, 2>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 2>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 2>
|
|
(const coeff[], const real[], real, real, real);
|
|
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<true, SphericalEngine::FULL, 3>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<false, SphericalEngine::FULL, 3>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 3>
|
|
(const coeff[], const real[], real, real, real);
|
|
template CircularEngine GEOGRAPHICLIB_EXPORT
|
|
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 3>
|
|
(const coeff[], const real[], real, real, real);
|
|
/// \endcond
|
|
|
|
} // namespace GeographicLib
|