void mexFunction(const int nlhs, mxArray *plhs[], const int nrhs, const mxArray *prhs[]) {
double u, scalFactor;
ClusterSetCPP<double> HBar;
ClusterSetCPP<double> dHBardu;//The first derivatives
ClusterSetCPP<double> d2HBardu2;//The second derivatives
size_t M, numH, i;
mxArray *CSRetVal;
mxArray *clusterElsMATLAB,*clusterSizesMATLAB, *offsetArrayMATLAB;
if(nrhs!=3){
mexErrMsgTxt("Incorrect number of inputs.");
return;
}
u=getDoubleFromMatlab(prhs[0]);
M=getSizeTFromMatlab(prhs[1]);
scalFactor=getDoubleFromMatlab(prhs[2]);
if(M<3) {
mexErrMsgTxt("The maximum order should be at least 3.");
return;
}
numH=(M+1)*(M+2)/2;
//Allocate space for the results.
clusterElsMATLAB=mxCreateDoubleMatrix(numH,1,mxREAL);
clusterSizesMATLAB=allocUnsignedSizeMatInMatlab(M+1,1);
offsetArrayMATLAB=allocUnsignedSizeMatInMatlab(M+1,1);
HBar.numClust=M+1;
HBar.totalNumEl=numH;
HBar.clusterEls=reinterpret_cast<double*>(mxGetData(clusterElsMATLAB));
HBar.offsetArray=reinterpret_cast<size_t*>(mxGetData(offsetArrayMATLAB));
HBar.clusterSizes=reinterpret_cast<size_t*>(mxGetData(clusterSizesMATLAB));
//Initialize the offset array and cluster sizes.
HBar.offsetArray[0]=0;
HBar.clusterSizes[0]=1;
for(i=1;i<=M;i++){
HBar.clusterSizes[i]=i+1;
HBar.offsetArray[i]=HBar.offsetArray[i-1]+HBar.clusterSizes[i-1];
}
normHelmHoltzCPP(HBar,u,scalFactor);
//Set the first return value
mexCallMATLAB(1,&CSRetVal,0, 0, "ClusterSet");
mxSetProperty(CSRetVal,0,"clusterEls",clusterElsMATLAB);
mxSetProperty(CSRetVal,0,"clusterSizes",clusterSizesMATLAB);
mxSetProperty(CSRetVal,0,"offsetArray",offsetArrayMATLAB);
plhs[0]=CSRetVal;
if(nlhs>1) {//Compute the first derivatives, if they are desired.
mxArray *clusterEls1stDerivMATLAB=mxCreateDoubleMatrix(numH,1,mxREAL);
dHBardu.numClust=M+1;
dHBardu.totalNumEl=numH;
dHBardu.clusterEls=reinterpret_cast<double*>(mxGetData(clusterEls1stDerivMATLAB));
dHBardu.offsetArray=reinterpret_cast<size_t*>(mxGetData(offsetArrayMATLAB));
dHBardu.clusterSizes=reinterpret_cast<size_t*>(mxGetData(clusterSizesMATLAB));
normHelmHoltzDerivCPP(dHBardu,HBar);
//Set the second return value
mexCallMATLAB(1,&CSRetVal,0, 0, "ClusterSet");
mxSetProperty(CSRetVal,0,"clusterEls",clusterEls1stDerivMATLAB);
mxSetProperty(CSRetVal,0,"clusterSizes",clusterSizesMATLAB);
mxSetProperty(CSRetVal,0,"offsetArray",offsetArrayMATLAB);
plhs[1]=CSRetVal;
mxDestroyArray(clusterEls1stDerivMATLAB);
}
if(nlhs>2) {//Compute the second derivatives if they are desired.
mxArray *clusterEls2ndDerivMATLAB=mxCreateDoubleMatrix(numH,1,mxREAL);
d2HBardu2.numClust=M+1;
d2HBardu2.totalNumEl=numH;
d2HBardu2.clusterEls=reinterpret_cast<double*>(mxGetData(clusterEls2ndDerivMATLAB));
d2HBardu2.offsetArray=reinterpret_cast<size_t*>(mxGetData(offsetArrayMATLAB));
d2HBardu2.clusterSizes=reinterpret_cast<size_t*>(mxGetData(clusterSizesMATLAB));
normHelmHoltzDeriv2CPP(d2HBardu2,HBar);
//Set the third return value
mexCallMATLAB(1,&CSRetVal,0, 0, "ClusterSet");
mxSetProperty(CSRetVal,0,"clusterEls",clusterEls2ndDerivMATLAB);
mxSetProperty(CSRetVal,0,"clusterSizes",clusterSizesMATLAB);
mxSetProperty(CSRetVal,0,"offsetArray",offsetArrayMATLAB);
plhs[2]=CSRetVal;
mxDestroyArray(clusterEls2ndDerivMATLAB);
}
//Free the buffers. The mxSetProperty command copied the data.
mxDestroyArray(clusterElsMATLAB);
mxDestroyArray(clusterSizesMATLAB);
mxDestroyArray(offsetArrayMATLAB);
}
void spherHarmonicEvalCPP(double *V, double *gradV, const ClusterSetCPP<double> &C,const ClusterSetCPP<double> &S, const double *point, const size_t numPoints, const double a, const double c, const double scalFactor) {
//If a NULL pointer is passed for gradV, then it is assumed that the
//gradient is not desired. Otherwise, a pointer to a buffer for 3
//doubles should be passed.
double temp, r, lambda, *nCoeff;
const size_t M=C.numClust-1;
const double pi = 2*acos(0.0);
size_t n,m,curPoint;
double nf,mf;
double rPrev,thetaPrev;
ClusterSetCPP<double> FuncVals;
ClusterSetCPP<double> FuncDerivs;
//These are to store sin(m*lambda) and cos(m*lambda) for m=0->M.
double *SinVec,*CosVec;//This are each length C.numClust.
//These are never used at the same time as SinVec andCosVec and are the
//same size, so they will point to the same memory.
double *rm,*im;
//These hold values for the modified forward row algorithm that stay
//constant for points with the same range and latitude but different
//longitudes.
double *XC,*XS;
//These values are only used if gradV!=NULL. There are initialized to
//NULL here to suppress a warning if compiled using
//-Wconditional-uninitialized
double *XCdr=NULL;
double *XSdr=NULL;
double *XCdTheta=NULL;
double *XSdTheta=NULL;
//A big chunk of memory will be allocated into a single buffer and
//split between the variables that need it. That is faster than
//allocating a bunch of small buffers, and all of the variables are of
//the same type.
double *buffer;
//Initialize the ClusterSet classes for the coefficients. The space
//for the elements will be allocated shortly.
FuncVals.numClust=C.numClust;
FuncVals.totalNumEl=C.totalNumEl;
FuncVals.offsetArray=C.offsetArray;
FuncVals.clusterSizes=C.clusterSizes;
FuncDerivs.numClust=C.numClust;
FuncDerivs.totalNumEl=C.totalNumEl;
FuncDerivs.offsetArray=C.offsetArray;
FuncDerivs.clusterSizes=C.clusterSizes;
//Allocate the buffer and partition it between variables.
if(gradV==NULL){
buffer = new double[C.totalNumEl+5*C.numClust];
}else{
buffer = new double[2*C.totalNumEl+9*C.numClust];
}
{
double *tempPtr=buffer;
//This stores all of the powers of a/r needed for the sum, regardless
//of which algorithm is used.
nCoeff=tempPtr;
tempPtr+=C.numClust;
//The sin and cosine values use the same memory as the im and rm
//values, because only one will be used depending on which
//algorithm is executed.
SinVec=tempPtr;
rm=SinVec;
tempPtr+=C.numClust;
CosVec=tempPtr;
im=CosVec;
tempPtr+=C.numClust;
FuncVals.clusterEls=tempPtr;
tempPtr+=C.totalNumEl;
XC=tempPtr;
tempPtr+=C.numClust;
XS=tempPtr;
if(gradV!=NULL) {
tempPtr+=C.numClust;
FuncDerivs.clusterEls=tempPtr;
tempPtr+=C.numClust;
XCdr=tempPtr;
tempPtr+=C.numClust;
XSdr=tempPtr;
tempPtr+=C.numClust;
XCdTheta=tempPtr;
tempPtr+=C.numClust;
XSdTheta=tempPtr;
}
}
nCoeff[0]=1;
rPrev=std::numeric_limits<double>::infinity();
thetaPrev=std::numeric_limits<double>::infinity();
for(curPoint=0;curPoint<numPoints;curPoint++) {
double thetaCur;
bool rChanged;
bool thetaChanged;
r=point[0+3*curPoint];
lambda=point[1+3*curPoint];
thetaCur=point[2+3*curPoint];
rChanged=rPrev!=r;
thetaChanged=thetaCur!=thetaPrev;
rPrev=r;
thetaPrev=thetaCur;
if(rChanged) {
temp=a/r;
for(n=1;n<=M;n++) {
nCoeff[n]=nCoeff[n-1]*temp;
}
}
if(fabs(thetaCur)<88*pi/180||gradV==NULL) {
//At latitudes that are not near the poles, the algorithm of Holmes and
//Featherstone is used. It can not be used for the gradient near the
//poles, because of the singularity of the spherical coordinate system.
double u, theta;
//Compute the sine and cosine terms
//Explicitely set the first two terms.
SinVec[0]=0;
CosVec[0]=1;
SinVec[1]=sin(lambda);
CosVec[1]=cos(lambda);
//Use a double angle identity to get the second order term.
SinVec[2]=2*SinVec[1]*CosVec[1];
CosVec[2]=1-2*SinVec[1]*SinVec[1];
//Use a two-part recursion for the rest of the terms.
for(m=3;m<=M;m++){
SinVec[m]=2*CosVec[1]*SinVec[m-1]-SinVec[m-2];
CosVec[m]=2*CosVec[1]*CosVec[m-1]-CosVec[m-2];
}
//The spherical coordinate system used in ellips2Sphere uses azimuth
//and elevation (latitude). However the formulae for spherical harmonic
//synthesis in the Holmes and Featherstone paper use, pi/2-elevation
//(colatitude). Thus, the point must be transformed.
theta=pi/2-thetaCur;
u=sin(theta);
if(thetaChanged) {
//Get the associated Legendre function ratios.
NALegendreCosRatCPP(FuncVals, theta, scalFactor);
//Get the derivatives of the ratios if the gradient is desired.
if(gradV!=NULL) {
NALegendreCosRatDerivCPP(FuncDerivs,FuncVals,theta);
}
}
//Evaluate Equation 7 from the Holmes and Featherstone paper.
if(rChanged||thetaChanged) {
//Zero the arrays.
memset(XC,0,sizeof(double)*C.numClust);
memset(XS,0,sizeof(double)*C.numClust);
//Compute the X coefficients for the sum
for(m=0;m<=M;m++) {
for(n=m;n<=M;n++) {
XC[m]+=nCoeff[n]*C[n][m]*FuncVals[n][m];
XS[m]+=nCoeff[n]*S[n][m]*FuncVals[n][m];
}
}
}
//Use Horner's method to compute V.
V[curPoint]=0;
m=M+1;
do {
m--;
V[curPoint]=V[curPoint]*u+XC[m]*CosVec[m]+XS[m]*SinVec[m];
} while(m>0);
//Multiple by the constant in front of the sum and get rid of the
//scale factor.
V[curPoint]=(c/r)*V[curPoint]/scalFactor;
//Compute the gradient, if it is desired.
if(gradV!=NULL) {
double J[9];
double dVdr=0;
double dVdLambda=0;
double dVdTheta=0;
if(rChanged||thetaChanged) {
memset(XCdr,0,sizeof(double)*C.numClust);
memset(XSdr,0,sizeof(double)*C.numClust);
memset(XCdTheta,0,sizeof(double)*C.numClust);
memset(XSdTheta,0,sizeof(double)*C.numClust);
//Evaluate Equation 7 from the Holmes and Featherstone paper.
mf=0;
for(m=0;m<=M;m++) {
nf=mf;
for(n=m;n<=M;n++) {
double CScal=nCoeff[n]*C[n][m];
double SScal=nCoeff[n]*S[n][m];
XCdr[m]+=(nf+1)*CScal*FuncVals[n][m];
XSdr[m]+=(nf+1)*SScal*FuncVals[n][m];
XCdTheta[m]+=CScal*FuncDerivs[n][m];
XSdTheta[m]+=SScal*FuncDerivs[n][m];
nf++;
}
mf++;
}
}
//Use Horner's method to compute the partials.
m=M+1;
mf=static_cast<double>(m);
do {
m--;
mf--;
dVdr=dVdr*u+XCdr[m]*CosVec[m]+XSdr[m]*SinVec[m];
dVdLambda=dVdLambda*u+mf*(-XC[m]*SinVec[m]+XS[m]*CosVec[m]);
dVdTheta=dVdTheta*u+XCdTheta[m]*CosVec[m]+XSdTheta[m]*SinVec[m];
} while(m>0);
dVdr=-(c/(r*r))*dVdr/scalFactor;
dVdLambda=(c/r)*dVdLambda/scalFactor;
//The minus sign is because the input coordinate was with respect
//to latitude, not the co-latitude that the NALegendreCosRat
//function uses.
dVdTheta=-(c/r)*dVdTheta/scalFactor;
calcSpherJacobCPP(J, point+3*curPoint,0);
//Now, multiply the transpose of the Jacobian Matrix by the
//vector of [dVdr;dVdLambda;dVdTheta]
gradV[0+3*curPoint]=dVdr*J[0]+dVdLambda*J[1]+dVdTheta*J[2];
gradV[1+3*curPoint]=dVdr*J[3]+dVdLambda*J[4]+dVdTheta*J[5];
gradV[2+3*curPoint]=dVdr*J[6]+dVdLambda*J[7]+dVdTheta*J[8];
}
} else {
//At latitudes that are near the poles, the non-singular algorithm of
//Pines using the fully normalized Helmholtz equations from Fantino and
//Casotto is used. The algorithm has been slightly modified so that the
//c/r term is out front and the fully normalized Helmholtz polynomials
//can be scaled. Also, lumped coefficients are not used. The Pines
//algorithm is generally slower than the algorithm of Holmes and
//Featherstone and it is suffers a loss of precision near the equator.
//Thus, the Pines algorithm is only used near the poles where the other
//algorithm has issues with a singularity.
double s,t,u, CartPoint[3];
spher2CartCPP(CartPoint,point+3*curPoint,0);
//Get the direction cosines used by Pines' algorithm.
s=CartPoint[0]/r;
t=CartPoint[1]/r;
u=CartPoint[2]/r;
//Compute the fully normalized Helmholtz polynomials.
if(thetaChanged) {
normHelmHoltzCPP(FuncVals,u, scalFactor);
thetaPrev=thetaCur;
}
//Recursively compute the rm and im terms for the sums.
rm[0]=1;
im[0]=0;
for(m=1;m<=M;m++) {
//These are equation 49 in the Fantino and Casotto paper.
rm[m]=s*rm[m-1]-t*im[m-1];
im[m]=s*im[m-1]+t*rm[m-1];
}
//Perform the sum for the potential from Equation 44 in the
//Fantino and Casotto paper.
V[curPoint]=0;
for(n=0;n<=M;n++) {
double innerTerm=0;
for(m=0;m<=n;m++) {
innerTerm+=(C[n][m]*rm[m]+S[n][m]*im[m])*FuncVals[n][m];
}
V[curPoint]+=nCoeff[n]*innerTerm;
}
V[curPoint]=(c/r)*V[curPoint]/scalFactor;
//Compute the gradient.
if(gradV!=NULL) {
double a1=0;
double a2=0;
double a3=0;
double a4=0;
normHelmHoltzDerivCPP(FuncDerivs,FuncVals);
//The equations in these loops are from Table 10.
nf=0.0;
for(n=0;n<=M;n++) {
double a1Loop=0;
double a2Loop=0;
double a3Loop;
double a4Loop;
double CProdMN;
double HVal;
double dHVal;
double Lmn;
//The m=0 case only applies to a3 and a4.
m=0;
mf=0.0;
HVal=FuncVals[n][m];
dHVal=FuncDerivs[n][m];
CProdMN=C[n][m]*rm[m]+S[n][m]*im[m];
a3Loop=CProdMN*dHVal;
Lmn=(nf+mf+1)*HVal+u*dHVal;//Defined in Table 14.
a4Loop=-CProdMN*Lmn;
mf=1.0;
for(m=1;m<=n;m++) {
HVal=FuncVals[n][m];
dHVal=FuncDerivs[n][m];
a1Loop+=mf*(C[n][m]*rm[m-1]+S[n][m]*im[m-1])*HVal;
a2Loop+=mf*(S[n][m]*rm[m-1]-C[n][m]*im[m-1])*HVal;
CProdMN=C[n][m]*rm[m]+S[n][m]*im[m];
a3Loop+=CProdMN*dHVal;
Lmn=(nf+mf+1)*HVal+u*dHVal;
a4Loop-=CProdMN*Lmn;
mf++;
}
a1+=nCoeff[n]*a1Loop;
a2+=nCoeff[n]*a2Loop;
a3+=nCoeff[n]*a3Loop;
a4+=nCoeff[n]*a4Loop;
nf++;
}
//These are equation 70. However, an additional 1/r term has been added,
//which the original paper omitted when going from Equation 68 to 70.
{
temp=c/(r*r);
gradV[0+3*curPoint]=temp*(a1+s*a4)/scalFactor;
gradV[1+3*curPoint]=temp*(a2+t*a4)/scalFactor;
gradV[2+3*curPoint]=temp*(a3+u*a4)/scalFactor;
}
}
}
}
delete[] buffer;
}
