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+/*****
+ * runarray.in
+ *
+ * Runtime functions for array operations.
+ *
+ *****/
+
+pair => primPair()
+triple => primTriple()
+boolarray* => booleanArray()
+Intarray* => IntArray()
+Intarray2* => IntArray2()
+realarray* => realArray()
+realarray2* => realArray2()
+pairarray* => pairArray()
+triplearray2* => tripleArray2()
+callableReal* => realRealFunction()
+
+
+#include "array.h"
+#include "arrayop.h"
+#include "triple.h"
+#include "path3.h"
+#include "Delaunay.h"
+
+#ifdef HAVE_LIBFFTW3
+#include "fftw++.h"
+#endif
+
+using namespace camp;
+using namespace vm;
+
+typedef array boolarray;
+typedef array Intarray;
+typedef array Intarray2;
+typedef array realarray;
+typedef array realarray2;
+typedef array pairarray;
+typedef array triplearray2;
+
+using types::booleanArray;
+using types::IntArray;
+using types::IntArray2;
+using types::realArray;
+using types::realArray2;
+using types::pairArray;
+using types::tripleArray2;
+
+typedef callable callableReal;
+
+void outOfBounds(const char *op, size_t len, Int n)
+{
+ ostringstream buf;
+ buf << op << " array of length " << len << " with out-of-bounds index " << n;
+ error(buf);
+}
+
+inline item& arrayRead(array *a, Int n)
+{
+ size_t len=checkArray(a);
+ bool cyclic=a->cyclic();
+ if(cyclic && len > 0) n=imod(n,len);
+ else if(n < 0 || n >= (Int) len) outOfBounds("reading",len,n);
+ return (*a)[(unsigned) n];
+}
+
+// Helper function to create deep arrays.
+static array* deepArray(Int depth, Int *dims)
+{
+ assert(depth > 0);
+
+ if (depth == 1) {
+ return new array(dims[0]);
+ } else {
+ Int length = dims[0];
+ depth--; dims++;
+
+ array *a = new array(length);
+
+ for (Int index = 0; index < length; index++) {
+ (*a)[index] = deepArray(depth, dims);
+ }
+ return a;
+ }
+}
+
+namespace run {
+array *Identity(Int n)
+{
+ size_t N=(size_t) n;
+ array *c=new array(N);
+ for(size_t i=0; i < N; ++i) {
+ array *ci=new array(N);
+ (*c)[i]=ci;
+ for(size_t j=0; j < N; ++j)
+ (*ci)[j]=0.0;
+ (*ci)[i]=1.0;
+ }
+ return c;
+}
+}
+
+static const char *incommensurate="Incommensurate matrices";
+static const char *singular="Singular matrix";
+static size_t *pivot,*Row,*Col;
+
+namespace run {
+
+array *copyArray(array *a)
+{
+ size_t size=checkArray(a);
+ array *c=new array(size);
+ for(size_t i=0; i < size; i++)
+ (*c)[i]=(*a)[i];
+ return c;
+}
+
+inline size_t checkdimension(const array *a, size_t dim)
+{
+ size_t size=checkArray(a);
+ if(dim && size != dim) {
+ ostringstream buf;
+ buf << "array of length " << dim << " expected";
+ error(buf);
+ }
+ return size;
+}
+
+double *copyArrayC(const array *a, size_t dim)
+{
+ size_t size=checkdimension(a,dim);
+ double *c=new double[size];
+ for(size_t i=0; i < size; i++)
+ c[i]=read<double>(a,i);
+ return c;
+}
+
+triple *copyTripleArrayC(const array *a, size_t dim)
+{
+ size_t size=checkdimension(a,dim);
+ triple *c=new triple[size];
+ for(size_t i=0; i < size; i++)
+ c[i]=read<triple>(a,i);
+ return c;
+}
+
+array *copyArray2(array *a)
+{
+ size_t size=checkArray(a);
+ array *c=new array(size);
+ for(size_t i=0; i < size; i++) {
+ array *ai=read<array*>(a,i);
+ size_t aisize=checkArray(ai);
+ array *ci=new array(aisize);
+ (*c)[i]=ci;
+ for(size_t j=0; j < aisize; j++)
+ (*ci)[j]=(*ai)[j];
+ }
+ return c;
+}
+
+array *copyArray3(array *a)
+{
+ size_t size=checkArray(a);
+ array *c=new array(size);
+ for(size_t i=0; i < size; i++) {
+ array *ai=read<array*>(a,i);
+ size_t aisize=checkArray(ai);
+ array *ci=new array(aisize);
+ (*c)[i]=ci;
+ for(size_t j=0; j < aisize; j++) {
+ array *aij=read<array*>(ai,j);
+ size_t aijsize=checkArray(aij);
+ array *cij=new array(aijsize);
+ (*ci)[j]=cij;
+ for(size_t k=0; k < aijsize; k++)
+ (*cij)[k]=(*aij)[k];
+ }
+ }
+ return c;
+}
+
+double *copyArray2C(const array *a, bool square, size_t dim2)
+{
+ size_t n=checkArray(a);
+ size_t m=(square || n == 0) ? n : checkArray(read<array*>(a,0));
+ if(n > 0 && dim2 && m != dim2) {
+ ostringstream buf;
+ buf << "second matrix dimension must be " << dim2;
+ error(buf);
+ }
+
+ double *c=new double[n*m];
+ for(size_t i=0; i < n; i++) {
+ array *ai=read<array*>(a,i);
+ size_t aisize=checkArray(ai);
+ if(aisize == m) {
+ double *ci=c+i*m;
+ for(size_t j=0; j < m; j++)
+ ci[j]=read<double>(ai,j);
+ } else
+ error(square ? "matrix must be square" : "matrix must be rectangular");
+ }
+ return c;
+}
+
+triple *copyTripleArray2C(const array *a, bool square, size_t dim2)
+{
+ size_t n=checkArray(a);
+ size_t m=(square || n == 0) ? n : checkArray(read<array*>(a,0));
+ if(n > 0 && dim2 && m != dim2) {
+ ostringstream buf;
+ buf << "second matrix dimension must be " << dim2;
+ error(buf);
+ }
+
+ triple *c=new triple[n*m];
+ for(size_t i=0; i < n; i++) {
+ array *ai=read<array*>(a,i);
+ size_t aisize=checkArray(ai);
+ if(aisize == m) {
+ triple *ci=c+i*m;
+ for(size_t j=0; j < m; j++)
+ ci[j]=read<triple>(ai,j);
+ } else
+ error(square ? "matrix must be square" : "matrix must be rectangular");
+ }
+ return c;
+}
+
+double *copyTripleArray2Components(array *a, bool square, size_t dim2)
+{
+ size_t n=checkArray(a);
+ size_t m=(square || n == 0) ? n : checkArray(read<array*>(a,0));
+ if(n > 0 && dim2 && m != dim2) {
+ ostringstream buf;
+ buf << "second matrix dimension must be " << dim2;
+ error(buf);
+ }
+
+ size_t nm=n*m;
+ double *cx=new double[3*nm];
+ double *cy=cx+nm;
+ double *cz=cx+2*nm;
+ for(size_t i=0; i < n; i++) {
+ array *ai=read<array*>(a,i);
+ size_t aisize=checkArray(ai);
+ if(aisize == m) {
+ double *xi=cx+i*m;
+ double *yi=cy+i*m;
+ double *zi=cz+i*m;
+ for(size_t j=0; j < m; j++) {
+ triple v=read<triple>(ai,j);
+ xi[j]=v.getx();
+ yi[j]=v.gety();
+ zi[j]=v.getz();
+ }
+ } else
+ error(square ? "matrix must be square" : "matrix must be rectangular");
+ }
+ return cx;
+}
+
+triple operator *(const array& t, const triple& v)
+{
+ size_t n=checkArray(&t);
+ if(n != 4) error(incommensurate);
+ array *t0=read<array*>(t,0);
+ array *t1=read<array*>(t,1);
+ array *t2=read<array*>(t,2);
+ array *t3=read<array*>(t,3);
+
+ if(checkArray(t0) != 4 || checkArray(t1) != 4 ||
+ checkArray(t2) != 4 || checkArray(t3) != 4)
+ error(incommensurate);
+
+ double x=v.getx();
+ double y=v.gety();
+ double z=v.getz();
+
+ double f=read<real>(t3,0)*x+read<real>(t3,1)*y+read<real>(t3,2)*z+
+ read<real>(t3,3);
+ if(f == 0.0) run::dividebyzero();
+ f=1.0/f;
+
+ return triple((read<real>(t0,0)*x+read<real>(t0,1)*y+read<real>(t0,2)*z+
+ read<real>(t0,3))*f,
+ (read<real>(t1,0)*x+read<real>(t1,1)*y+read<real>(t1,2)*z+
+ read<real>(t1,3))*f,
+ (read<real>(t2,0)*x+read<real>(t2,1)*y+read<real>(t2,2)*z+
+ read<real>(t2,3))*f);
+}
+
+triple multshiftless(const array& t, const triple& v)
+{
+ size_t n=checkArray(&t);
+ if(n != 4) error(incommensurate);
+ array *t0=read<array*>(t,0);
+ array *t1=read<array*>(t,1);
+ array *t2=read<array*>(t,2);
+ array *t3=read<array*>(t,3);
+
+ if(checkArray(t0) != 4 || checkArray(t1) != 4 ||
+ checkArray(t2) != 4 || checkArray(t3) != 4)
+ error(incommensurate);
+
+ double x=v.getx();
+ double y=v.gety();
+ double z=v.getz();
+
+ double f=read<real>(t3,0)*x+read<real>(t3,1)*y+read<real>(t3,2)*z+
+ read<real>(t3,3);
+ if(f == 0.0) run::dividebyzero();
+ f=1.0/f;
+
+ return triple((read<real>(t0,0)*x+read<real>(t0,1)*y+read<real>(t0,2)*z)*f,
+ (read<real>(t1,0)*x+read<real>(t1,1)*y+read<real>(t1,2)*z)*f,
+ (read<real>(t2,0)*x+read<real>(t2,1)*y+read<real>(t2,2)*z)*f);
+}
+
+double norm(double *a, size_t n)
+{
+ if(n == 0) return 0.0;
+ double M=fabs(a[0]);
+ for(size_t i=1; i < n; ++i)
+ M=::max(M,fabs(a[i]));
+ return M;
+}
+
+double norm(triple *a, size_t n)
+{
+ if(n == 0) return 0.0;
+ double M=a[0].abs2();
+ for(size_t i=1; i < n; ++i)
+ M=::max(M,a[i].abs2());
+ return sqrt(M);
+}
+
+}
+
+static inline void inverseAllocate(size_t n)
+{
+ pivot=new size_t[n];
+ Row=new size_t[n];
+ Col=new size_t[n];
+}
+
+static inline void inverseDeallocate()
+{
+ delete[] pivot;
+ delete[] Row;
+ delete[] Col;
+}
+
+callable *Func;
+stack *FuncStack;
+double wrapFunction(double x)
+{
+ FuncStack->push(x);
+ Func->call(FuncStack);
+ return pop<double>(FuncStack);
+}
+
+callable *compareFunc;
+bool compareFunction(const vm::item& i, const vm::item& j)
+{
+ FuncStack->push(i);
+ FuncStack->push(j);
+ compareFunc->call(FuncStack);
+ return pop<bool>(FuncStack);
+}
+
+void checkSquare(array *a)
+{
+ size_t n=checkArray(a);
+ for(size_t i=0; i < n; i++)
+ if(checkArray(read<array*>(a,i)) != n)
+ error("matrix a must be square");
+}
+
+// Crout's algorithm for computing the LU decomposition of a square matrix.
+// cf. routine ludcmp (Press et al., Numerical Recipes, 1991).
+Int LUdecompose(double *a, size_t n, size_t* index, bool warn=true)
+{
+ double *vv=new double[n];
+ Int swap=1;
+ for(size_t i=0; i < n; ++i) {
+ double big=0.0;
+ double *ai=a+i*n;
+ for(size_t j=0; j < n; ++j) {
+ double temp=fabs(ai[j]);
+ if(temp > big) big=temp;
+ }
+ if(big == 0.0) {
+ delete[] vv;
+ if(warn) error(singular);
+ else return 0;
+ }
+ vv[i]=1.0/big;
+ }
+ for(size_t j=0; j < n; ++j) {
+ for(size_t i=0; i < j; ++i) {
+ double *ai=a+i*n;
+ double sum=ai[j];
+ for(size_t k=0; k < i; ++k) {
+ sum -= ai[k]*a[k*n+j];
+ }
+ ai[j]=sum;
+ }
+ double big=0.0;
+ size_t imax=j;
+ for(size_t i=j; i < n; ++i) {
+ double *ai=a+i*n;
+ double sum=ai[j];
+ for(size_t k=0; k < j; ++k)
+ sum -= ai[k]*a[k*n+j];
+ ai[j]=sum;
+ double temp=vv[i]*fabs(sum);
+ if(temp >= big) {
+ big=temp;
+ imax=i;
+ }
+ }
+ double *aj=a+j*n;
+ double *aimax=a+imax*n;
+ if(j != imax) {
+ for(size_t k=0; k < n; ++k) {
+ double temp=aimax[k];
+ aimax[k]=aj[k];
+ aj[k]=temp;
+ }
+ swap *= -1;
+ vv[imax]=vv[j];
+ }
+ if(index)
+ index[j]=imax;
+ if(j != n) {
+ double denom=aj[j];
+ if(denom == 0.0) {
+ delete[] vv;
+ if(warn) error(singular);
+ else return 0;
+ }
+ for(size_t i=j+1; i < n; ++i)
+ a[i*n+j] /= denom;
+ }
+ }
+ delete[] vv;
+ return swap;
+}
+
+namespace run {
+void dividebyzero(size_t i)
+{
+ ostringstream buf;
+ if(i > 0) buf << "array element " << i << ": ";
+ buf << "Divide by zero";
+ error(buf);
+}
+
+void integeroverflow(size_t i)
+{
+ ostringstream buf;
+ if(i > 0) buf << "array element " << i << ": ";
+ buf << "Integer overflow";
+ error(buf);
+}
+}
+
+// Autogenerated routines:
+
+
+// Create an empty array.
+array* :emptyArray()
+{
+ return new array(0);
+}
+
+// Create a new array (technically a vector).
+// This array will be multidimensional. First the number of dimensions
+// is popped off the stack, followed by each dimension in reverse order.
+// The array itself is technically a one dimensional array of one
+// dimension arrays and so on.
+array* :newDeepArray(Int depth)
+{
+ assert(depth > 0);
+
+ Int *dims = new Int[depth];
+
+ for (Int index = depth-1; index >= 0; index--) {
+ Int i=pop<Int>(Stack);
+ if(i < 0) error("cannot create a negative length array");
+ dims[index]=i;
+ }
+
+ array *a=deepArray(depth, dims);
+ delete[] dims;
+ return a;
+}
+
+// Creates an array with elements already specified. First, the number
+// of elements is popped off the stack, followed by each element in
+// reverse order.
+array* :newInitializedArray(Int n)
+{
+ assert(n >= 0);
+
+ array *a = new array(n);
+
+ for (Int index = n-1; index >= 0; index--)
+ (*a)[index] = pop(Stack);
+
+ return a;
+}
+
+// Similar to newInitializedArray, but after the n elements, append another
+// array to it.
+array* :newAppendedArray(array* tail, Int n)
+{
+ assert(n >= 0);
+
+ array *a = new array(n);
+
+ for (Int index = n-1; index >= 0; index--)
+ (*a)[index] = pop(Stack);
+
+ copy(tail->begin(), tail->end(), back_inserter(*a));
+
+ return a;
+}
+
+// The function T[] array(int n, T value, int depth=0) produces a array of n
+// copies of x, where each copy is copied up to depth.
+array* :newDuplicateArray(Int n, item value, Int depth=Int_MAX)
+{
+ if(n < 0) error("cannot create a negative length array");
+ if(depth < 0) error("cannot copy to a negative depth");
+
+ return new array(n, value, depth);
+}
+
+// Read an element from an array. Checks for initialization & bounds.
+item :arrayRead(array *a, Int n)
+{
+ item& i=arrayRead(a,n);
+ if (i.empty()) {
+ ostringstream buf;
+ buf << "read uninitialized value from array at index " << n;
+ error(buf);
+ }
+ return i;
+}
+
+// Slice a substring from an array.
+item :arraySliceRead(array *a, Int left, Int right)
+{
+ checkArray(a);
+ return a->slice(left, right);
+}
+
+// Slice a substring from an array. This implements the cases a[i:] and a[:]
+// where the endpoint is not given, and assumed to be the length of the array.
+item :arraySliceReadToEnd(array *a, Int left)
+{
+ size_t len=checkArray(a);
+ return a->slice(left, (Int)len);
+}
+
+// Read an element from an array of arrays. Check bounds and initialize
+// as necessary.
+item :arrayArrayRead(array *a, Int n)
+{
+ item& i=arrayRead(a,n);
+ if (i.empty()) i=new array(0);
+ return i;
+}
+
+// Write an element to an array. Increase size if necessary.
+item :arrayWrite(item value, array *a, Int n)
+{
+ size_t len=checkArray(a);
+ bool cyclic=a->cyclic();
+ if(cyclic && len > 0) n=imod(n,len);
+ else {
+ if(cyclic) outOfBounds("writing cyclic",len,n);
+ if(n < 0) outOfBounds("writing",len,n);
+ if(len <= (size_t) n)
+ a->resize(n+1);
+ }
+ (*a)[n] = value;
+ return value;
+}
+
+array * :arraySliceWrite(array *src, array *dest, Int left, Int right)
+{
+ checkArray(src);
+ checkArray(dest);
+ dest->setSlice(left, right, src);
+ return src;
+}
+
+array * :arraySliceWriteToEnd(array *src, array *dest, Int left)
+{
+ checkArray(src);
+ size_t len=checkArray(dest);
+ dest->setSlice(left, (Int) len, src);
+ return src;
+}
+
+// Returns the length of an array.
+Int :arrayLength(array *a)
+{
+ return (Int) checkArray(a);
+}
+
+// Returns an array of integers representing the keys of the array.
+array * :arrayKeys(array *a)
+{
+ size_t size=checkArray(a);
+
+ array *keys=new array();
+ for (size_t i=0; i<size; ++i) {
+ item& cell = (*a)[i];
+ if (!cell.empty())
+ keys->push((Int)i);
+ }
+
+ return keys;
+}
+
+// Return the cyclic flag for an array.
+bool :arrayCyclicFlag(array *a)
+{
+ checkArray(a);
+ return a->cyclic();
+}
+
+bool :arraySetCyclicFlag(bool b, array *a)
+{
+ checkArray(a);
+ a->cyclic(b);
+ return b;
+}
+
+// Check to see if an array element is initialized.
+bool :arrayInitializedHelper(Int n, array *a)
+{
+ size_t len=checkArray(a);
+ bool cyclic=a->cyclic();
+ if(cyclic && len > 0) n=imod(n,len);
+ else if(n < 0 || n >= (Int) len) return false;
+ item&i=(*a)[(unsigned) n];
+ return !i.empty();
+}
+
+// Returns the initialize method for an array.
+callable* :arrayInitialized(array *a)
+{
+ return new thunk(new bfunc(arrayInitializedHelper),a);
+}
+
+// The helper function for the cyclic method that sets the cyclic flag.
+void :arrayCyclicHelper(bool b, array *a)
+{
+ checkArray(a);
+ a->cyclic(b);
+}
+
+// Set the cyclic flag for an array.
+callable* :arrayCyclic(array *a)
+{
+ return new thunk(new bfunc(arrayCyclicHelper),a);
+}
+
+// The helper function for the push method that does the actual operation.
+item :arrayPushHelper(item x, array *a)
+{
+ checkArray(a);
+ a->push(x);
+ return x;
+}
+
+// Returns the push method for an array.
+callable* :arrayPush(array *a)
+{
+ return new thunk(new bfunc(arrayPushHelper),a);
+}
+
+// The helper function for the append method that appends b to a.
+void :arrayAppendHelper(array *b, array *a)
+{
+ checkArray(a);
+ size_t size=checkArray(b);
+ for(size_t i=0; i < size; i++)
+ a->push((*b)[i]);
+}
+
+// Returns the append method for an array.
+callable* :arrayAppend(array *a)
+{
+ return new thunk(new bfunc(arrayAppendHelper),a);
+}
+
+// The helper function for the pop method.
+item :arrayPopHelper(array *a)
+{
+ size_t asize=checkArray(a);
+ if(asize == 0)
+ error("cannot pop element from empty array");
+ return a->pop();
+}
+
+// Returns the pop method for an array.
+callable* :arrayPop(array *a)
+{
+ return new thunk(new bfunc(arrayPopHelper),a);
+}
+
+// The helper function for the insert method.
+item :arrayInsertHelper(Int i, array *x, array *a)
+{
+ size_t asize=checkArray(a);
+ checkArray(x);
+ if(a->cyclic() && asize > 0) i=imod(i,asize);
+ if(i < 0 || i > (Int) asize)
+ outOfBounds("inserting",asize,i);
+ (*a).insert((*a).begin()+i,(*x).begin(),(*x).end());
+}
+
+// Returns the insert method for an array.
+callable* :arrayInsert(array *a)
+{
+ return new thunk(new bfunc(arrayInsertHelper),a);
+}
+
+// Returns the delete method for an array.
+callable* :arrayDelete(array *a)
+{
+ return new thunk(new bfunc(arrayDeleteHelper),a);
+}
+
+bool :arrayAlias(array *a, array *b)
+{
+ return a==b;
+}
+
+// Return array formed by indexing array a with elements of integer array b
+array* :arrayIntArray(array *a, array *b)
+{
+ size_t asize=checkArray(a);
+ size_t bsize=checkArray(b);
+ array *r=new array(bsize);
+ bool cyclic=a->cyclic();
+ for(size_t i=0; i < bsize; i++) {
+ Int index=read<Int>(b,i);
+ if(cyclic && asize > 0) index=imod(index,asize);
+ else
+ if(index < 0 || index >= (Int) asize)
+ outOfBounds("reading",asize,index);
+ (*r)[i]=(*a)[index];
+ }
+ return r;
+}
+
+// returns the complement of the integer array a in {0,2,...,n-1},
+// so that b[complement(a,b.length)] yields the complement of b[a].
+Intarray* complement(Intarray *a, Int n)
+{
+ size_t asize=checkArray(a);
+ array *r=new array(0);
+ bool *keep=new bool[n];
+ for(Int i=0; i < n; ++i) keep[i]=true;
+ for(size_t i=0; i < asize; ++i) {
+ Int j=read<Int>(a,i);
+ if(j >= 0 && j < n) keep[j]=false;
+ }
+ for(Int i=0; i < n; i++)
+ if(keep[i]) r->push(i);
+
+ delete[] keep;
+ return r;
+}
+
+// Generate the sequence {f(i) : i=0,1,...n-1} given a function f and integer n
+Intarray* :arraySequence(callable *f, Int n)
+{
+ if(n < 0) n=0;
+ array *a=new array(n);
+ for(Int i=0; i < n; ++i) {
+ Stack->push(i);
+ f->call(Stack);
+ (*a)[i]=pop(Stack);
+ }
+ return a;
+}
+
+// Return the array {0,1,...n-1}
+Intarray *sequence(Int n)
+{
+ if(n < 0) n=0;
+ array *a=new array(n);
+ for(Int i=0; i < n; ++i) {
+ (*a)[i]=i;
+ }
+ return a;
+}
+
+// Apply a function to each element of an array
+array* :arrayFunction(callable *f, array *a)
+{
+ size_t size=checkArray(a);
+ array *b=new array(size);
+ for(size_t i=0; i < size; ++i) {
+ Stack->push((*a)[i]);
+ f->call(Stack);
+ (*b)[i]=pop(Stack);
+ }
+ return b;
+}
+
+array* :arraySort(array *a, callable *f)
+{
+ array *c=copyArray(a);
+ compareFunc=f;
+ FuncStack=Stack;
+ stable_sort(c->begin(),c->end(),compareFunction);
+ return c;
+}
+
+bool all(boolarray *a)
+{
+ size_t size=checkArray(a);
+ bool c=true;
+ for(size_t i=0; i < size; i++)
+ if(!get<bool>((*a)[i])) {c=false; break;}
+ return c;
+}
+
+boolarray* !(boolarray* a)
+{
+ size_t size=checkArray(a);
+ array *c=new array(size);
+ for(size_t i=0; i < size; i++)
+ (*c)[i]=!read<bool>(a,i);
+ return c;
+}
+
+Int sum(boolarray *a)
+{
+ size_t size=checkArray(a);
+ Int sum=0;
+ for(size_t i=0; i < size; i++)
+ sum += read<bool>(a,i) ? 1 : 0;
+ return sum;
+}
+
+array* :arrayCopy(array *a)
+{
+ return copyArray(a);
+}
+
+array* :arrayConcat(array *a)
+{
+ // a is an array of arrays to be concatenated together.
+ // The signature is
+ // T[] concat(... T[][] a);
+
+ size_t numArgs=checkArray(a);
+ size_t resultSize=0;
+ for (size_t i=0; i < numArgs; ++i) {
+ resultSize += checkArray(a->read<array *>(i));
+ }
+
+ array *result=new array(resultSize);
+
+ size_t ri=0;
+ for (size_t i=0; i < numArgs; ++i) {
+ array *arg=a->read<array *>(i);
+ size_t size=checkArray(arg);
+
+ for (size_t j=0; j < size; ++j) {
+ (*result)[ri]=(*arg)[j];
+ ++ri;
+ }
+ }
+
+ return result;
+}
+
+array* :array2Copy(array *a)
+{
+ return copyArray2(a);
+}
+
+array* :array3Copy(array *a)
+{
+ return copyArray3(a);
+}
+
+array* :array2Transpose(array *a)
+{
+ size_t asize=checkArray(a);
+ array *c=new array(0);
+ for(size_t i=0; i < asize; i++) {
+ size_t ip=i+1;
+ array *ai=read<array*>(a,i);
+ size_t aisize=checkArray(ai);
+ size_t csize=checkArray(c);
+ if(csize < aisize) {
+ c->resize(aisize);
+ for(size_t j=csize; j < aisize; j++) {
+ (*c)[j]=new array(ip);
+ }
+ }
+ for(size_t j=0; j < aisize; j++) {
+ array *cj=read<array*>(c,j);
+ if(checkArray(cj) < ip) cj->resize(ip);
+ (*cj)[i]=(*ai)[j];
+ }
+ }
+ return c;
+}
+
+// a is a rectangular 3D array; perm is an Int array indicating the type of
+// permutation (021 or 120, etc; original is 012).
+// Transpose by sending respective members to the permutated locations:
+// return the array obtained by putting a[i][j][k] into position perm{ijk}.
+array* :array3Transpose(array *a, array *perm)
+{
+ const size_t DIM=3;
+
+ if(checkArray(perm) != DIM) {
+ ostringstream buf;
+ buf << "permutation array must have length " << DIM;
+ error(buf);
+ }
+
+ size_t* size=new size_t[DIM];
+ for(size_t i=0; i < DIM; ++i) size[i]=DIM;
+
+ for(size_t i=0; i < DIM; ++i) {
+ Int p=read<Int>(perm,i);
+ size_t P=(size_t) p;
+ if(p < 0 || P >= DIM) {
+ ostringstream buf;
+ buf << "permutation index out of range: " << p;
+ error(buf);
+ }
+ size[P]=P;
+ }
+
+ for(size_t i=0; i < DIM; ++i)
+ if(size[i] == DIM) error("permutation indices must be distinct");
+
+ static const char *rectangular=
+ "3D transpose implemented for rectangular matrices only";
+
+ size_t isize=size[0]=checkArray(a);
+ array *a0=read<array*>(a,0);
+ size[1]=checkArray(a0);
+ array *a00=read<array*>(a0,0);
+ size[2]=checkArray(a00);
+ for(size_t i=0; i < isize; i++) {
+ array *ai=read<array*>(a,i);
+ size_t jsize=checkArray(ai);
+ if(jsize != size[1]) error(rectangular);
+ for(size_t j=0; j < jsize; j++) {
+ array *aij=read<array*>(ai,j);
+ if(checkArray(aij) != size[2]) error(rectangular);
+ }
+ }
+
+ size_t perm0=(size_t) read<Int>(perm,0);
+ size_t perm1=(size_t) read<Int>(perm,1);
+ size_t perm2=(size_t) read<Int>(perm,2);
+
+ size_t sizep0=size[perm0];
+ size_t sizep1=size[perm1];
+ size_t sizep2=size[perm2];
+
+ array *c=new array(sizep0);
+ for(size_t i=0; i < sizep0; ++i) {
+ array *ci=new array(sizep1);
+ (*c)[i]=ci;
+ for(size_t j=0; j < sizep1; ++j) {
+ array *cij=new array(sizep2);
+ (*ci)[j]=cij;
+ }
+ }
+
+ size_t* i=new size_t[DIM];
+
+ for(i[0]=0; i[0] < size[0]; ++i[0]) {
+ array *a0=read<array*>(a,i[0]);
+ for(i[1]=0; i[1] < size[1]; ++i[1]) {
+ array *a1=read<array*>(a0,i[1]);
+ for(i[2]=0; i[2] < size[2]; ++i[2]) {
+ array *c0=read<array*>(c,i[perm0]);
+ array *c1=read<array*>(c0,i[perm1]);
+ (*c1)[i[perm2]]=read<real>(a1,i[2]);
+ }
+ }
+ }
+
+ delete [] i;
+ delete [] size;
+
+ return c;
+}
+
+// In a boolean array, find the index of the nth true value or -1 if not found
+// If n is negative, search backwards.
+Int find(boolarray *a, Int n=1)
+{
+ size_t size=checkArray(a);
+ Int j=-1;
+ if(n > 0)
+ for(size_t i=0; i < size; i++)
+ if(read<bool>(a,i)) {
+ n--; if(n == 0) {j=(Int) i; break;}
+ }
+ if(n < 0)
+ for(size_t i=size; i > 0;)
+ if(read<bool>(a,--i)) {
+ n++; if(n == 0) {j=(Int) i; break;}
+ }
+ return j;
+}
+
+// construct vector obtained by replacing those elements of b for which the
+// corresponding elements of a are false by the corresponding element of c.
+array* :arrayConditional(array *a, array *b, array *c)
+{
+ size_t size=checkArray(a);
+ array *r=new array(size);
+ if(b && c) {
+ checkArrays(a,b);
+ checkArrays(b,c);
+ for(size_t i=0; i < size; i++)
+ (*r)[i]=read<bool>(a,i) ? (*b)[i] : (*c)[i];
+ } else {
+ r->clear();
+ if(b) {
+ checkArrays(a,b);
+ for(size_t i=0; i < size; i++)
+ if(read<bool>(a,i)) r->push((*b)[i]);
+ } else if(c) {
+ checkArrays(a,c);
+ for(size_t i=0; i < size; i++)
+ if(!read<bool>(a,i)) r->push((*c)[i]);
+ }
+ }
+ return r;
+}
+
+// Return an n x n identity matrix.
+realarray2 *identity(Int n)
+{
+ return Identity(n);
+}
+
+// Return the diagonal matrix with diagonal entries given by a.
+realarray2* :diagonal(realarray *a)
+{
+ size_t n=checkArray(a);
+ array *c=new array(n);
+ for(size_t i=0; i < n; ++i) {
+ array *ci=new array(n);
+ (*c)[i]=ci;
+ for(size_t j=0; j < i; ++j)
+ (*ci)[j]=0.0;
+ (*ci)[i]=read<real>(a,i);
+ for(size_t j=i+1; j < n; ++j)
+ (*ci)[j]=0.0;
+ }
+ return c;
+}
+
+// Return the inverse of an n x n matrix a using Gauss-Jordan elimination.
+realarray2 *inverse(realarray2 *a)
+{
+ a=copyArray2(a);
+ size_t n=checkArray(a);
+ checkSquare(a);
+
+ inverseAllocate(n);
+
+ for(size_t i=0; i < n; i++)
+ pivot[i]=0;
+
+ size_t col=0, row=0;
+ // This is the main loop over the columns to be reduced.
+ for(size_t i=0; i < n; i++) {
+ real big=0.0;
+ // This is the outer loop of the search for a pivot element.
+ for(size_t j=0; j < n; j++) {
+ array *aj=read<array*>(a,j);
+ if(pivot[j] != 1) {
+ for(size_t k=0; k < n; k++) {
+ if(pivot[k] == 0) {
+ real temp=fabs(read<real>(aj,k));
+ if(temp >= big) {
+ big=temp;
+ row=j;
+ col=k;
+ }
+ } else if(pivot[k] > 1) {
+ inverseDeallocate();
+ error(singular);
+ }
+ }
+ }
+ }
+ ++(pivot[col]);
+
+ // Interchange rows, if needed, to put the pivot element on the diagonal.
+ array *acol=read<array*>(a,col);
+ if(row != col) {
+ array *arow=read<array*>(a,row);
+ for(size_t l=0; l < n; l++) {
+ real temp=read<real>(arow,l);
+ (*arow)[l]=read<real>(acol,l);
+ (*acol)[l]=temp;
+ }
+ }
+
+ Row[i]=row;
+ Col[i]=col;
+
+ // Divide the pivot row by the pivot element.
+ real denom=read<real>(acol,col);
+ if(denom == 0.0) {
+ inverseDeallocate();
+ error(singular);
+ }
+ real pivinv=1.0/denom;
+ (*acol)[col]=1.0;
+ for(size_t l=0; l < n; l++)
+ (*acol)[l]=read<real>(acol,l)*pivinv;
+
+ // Reduce all rows except for the pivoted one.
+ for(size_t k=0; k < n; k++) {
+ if(k != col) {
+ array *ak=read<array*>(a,k);
+ real akcol=read<real>(ak,col);
+ (*ak)[col]=0.0;
+ for(size_t l=0; l < n; l++)
+ (*ak)[l]=read<real>(ak,l)-read<real>(acol,l)*akcol;
+ }
+ }
+ }
+
+ // Unscramble the inverse matrix in view of the column interchanges.
+ for(size_t l=n; l > 0;) {
+ l--;
+ size_t r=Row[l];
+ size_t c=Col[l];
+ if(r != c) {
+ for(size_t k=0; k < n; k++) {
+ array *ak=read<array*>(a,k);
+ real temp=read<real>(ak,r);
+ (*ak)[r]=read<real>(ak,c);
+ (*ak)[c]=temp;
+ }
+ }
+ }
+ inverseDeallocate();
+ return a;
+}
+
+// Solve the linear equation ax=b by LU decomposition, returning the
+// solution x, where a is an n x n matrix and b is an array of length n.
+// If no solution exists, return an empty array.
+realarray *solve(realarray2 *a, realarray *b, bool warn=true)
+{
+ size_t n=checkArray(a);
+
+ if(n == 0) return new array(0);
+
+ size_t m=checkArray(b);
+ if(m != n) error(incommensurate);
+
+ real *A=copyArray2C(a);
+ size_t *index=new size_t[n];
+
+ if(LUdecompose(A,n,index,warn) == 0)
+ return new array(0);
+
+ array *x=new array(n);
+
+ real *B=copyArrayC(b);
+
+ for(size_t i=0; i < n; ++i) {
+ size_t ip=index[i];
+ real sum=B[ip];
+ B[ip]=B[i];
+ real *Ai=A+i*n;
+ for(size_t j=0; j < i; ++j)
+ sum -= Ai[j]*B[j];
+ B[i]=sum;
+ }
+
+ for(size_t i=n; i > 0;) {
+ --i;
+ real sum=B[i];
+ real *Ai=A+i*n;
+ for(size_t j=i+1; j < n; ++j)
+ sum -= Ai[j]*B[j];
+ B[i]=sum/Ai[i];
+ }
+
+ for(size_t i=0; i < n; ++i)
+ (*x)[i]=B[i];
+
+ delete[] index;
+ delete[] B;
+ delete[] A;
+
+ return x;
+}
+
+// Solve the linear equation ax=b by LU decomposition, returning the
+// solution x, where a is an n x n matrix and b is an n x m matrix.
+// If no solution exists, return an empty array.
+realarray2 *solve(realarray2 *a, realarray2 *b, bool warn=true)
+{
+ size_t n=checkArray(a);
+
+ if(n == 0) return new array(0);
+
+ if(checkArray(b) != n) error(incommensurate);
+ size_t m=checkArray(read<array*>(b,0));
+
+ real *A=copyArray2C(a);
+ real *B=copyArray2C(b,false);
+
+ size_t *index=new size_t[n];
+
+ if(LUdecompose(A,n,index,warn) == 0)
+ return new array(0);
+
+ array *x=new array(n);
+
+ for(size_t i=0; i < n; ++i) {
+ real *Ai=A+i*n;
+ real *Bi=B+i*m;
+ real *Bip=B+index[i]*m;
+ for(size_t k=0; k < m; ++k) {
+ real sum=Bip[k];
+ Bip[k]=Bi[k];
+ size_t jk=k;
+ for(size_t j=0; j < i; ++j, jk += m)
+ sum -= Ai[j]*B[jk];
+ Bi[k]=sum;
+ }
+ }
+
+ for(size_t i=n; i > 0;) {
+ --i;
+ real *Ai=A+i*n;
+ real *Bi=B+i*m;
+ for(size_t k=0; k < m; ++k) {
+ real sum=Bi[k];
+ size_t jk=(i+1)*m+k;
+ for(size_t j=i+1; j < n; ++j, jk += m)
+ sum -= Ai[j]*B[jk];
+ Bi[k]=sum/Ai[i];
+ }
+ }
+
+ for(size_t i=0; i < n; ++i) {
+ real *Bi=B+i*m;
+ array *xi=new array(m);
+ (*x)[i]=xi;
+ for(size_t j=0; j < m; ++j)
+ (*xi)[j]=Bi[j];
+ }
+
+ delete[] index;
+ delete[] B;
+ delete[] A;
+
+ return x;
+}
+
+// Compute the determinant of an n x n matrix.
+real determinant(realarray2 *a)
+{
+ real *A=copyArray2C(a);
+ size_t n=checkArray(a);
+
+ real det=LUdecompose(A,n,NULL,false);
+ size_t n1=n+1;
+ for(size_t i=0; i < n; ++i)
+ det *= A[i*n1];
+
+ delete[] A;
+
+ return det;
+}
+
+realarray *Operator *(realarray2 *a, realarray *b)
+{
+ size_t n=checkArray(a);
+ size_t m=checkArray(b);
+ array *c=new array(n);
+ real *B=copyArrayC(b);
+ for(size_t i=0; i < n; ++i) {
+ array *ai=read<array*>(a,i);
+ if(checkArray(ai) != m) error(incommensurate);
+ real sum=0.0;
+ for(size_t j=0; j < m; ++j)
+ sum += read<real>(ai,j)*B[j];
+ (*c)[i]=sum;
+ }
+ delete[] B;
+ return c;
+}
+
+realarray *Operator *(realarray *a, realarray2 *b)
+{
+ size_t n=checkArray(a);
+ if(n != checkArray(b)) error(incommensurate);
+ real *A=copyArrayC(a);
+
+ array **B=new array*[n];
+ array *bk=read<array *>(b,0);
+ B[0]=bk;
+ size_t m=bk->size();
+ for(size_t k=1; k < n; k++) {
+ array *bk=read<array *>(b,k);
+ if(bk->size() != m) error(incommensurate);
+ B[k]=bk;
+ }
+ array *c=new array(m);
+
+ for(size_t i=0; i < m; ++i) {
+ real sum=0.0;
+ for(size_t k=0; k < n; ++k)
+ sum += A[k]*read<real>(B[k],i);
+ (*c)[i]=sum;
+ }
+ delete[] B;
+ delete[] A;
+ return c;
+}
+
+realarray2 *Operator *(realarray2 *a, realarray2 *b)
+{
+ size_t n=checkArray(a);
+
+ size_t nb=checkArray(b);
+ size_t na0=n == 0 ? 0 : checkArray(read<array*>(a,0));
+ if(na0 != nb)
+ error(incommensurate);
+
+ size_t nb0=nb == 0 ? 0 : checkArray(read<array*>(b,0));
+
+ array *c=new array(n);
+
+ real *A=copyArray2C(a,false);
+ real *B=copyArray2C(b,false);
+
+ for(size_t i=0; i < n; ++i) {
+ real *Ai=A+i*nb;
+ array *ci=new array(nb0);
+ (*c)[i]=ci;
+ for(size_t j=0; j < nb0; ++j) {
+ real sum=0.0;
+ size_t kj=j;
+ for(size_t k=0; k < nb; ++k, kj += nb0)
+ sum += Ai[k]*B[kj];
+ (*ci)[j]=sum;
+ }
+ }
+
+ delete[] B;
+ delete[] A;
+
+ return c;
+}
+
+triple Operator *(realarray2 *t, triple v)
+{
+ return *t*v;
+}
+
+pair project(triple v, realarray2 *t)
+{
+ size_t n=checkArray(t);
+ if(n != 4) error(incommensurate);
+ array *t0=read<array*>(t,0);
+ array *t1=read<array*>(t,1);
+ array *t3=read<array*>(t,3);
+ if(checkArray(t0) != 4 || checkArray(t1) != 4 || checkArray(t3) != 4)
+ error(incommensurate);
+
+ real x=v.getx();
+ real y=v.gety();
+ real z=v.getz();
+
+ real f=read<real>(t3,0)*x+read<real>(t3,1)*y+read<real>(t3,2)*z+
+ read<real>(t3,3);
+ if(f == 0.0) dividebyzero();
+ f=1.0/f;
+
+ return pair((read<real>(t0,0)*x+read<real>(t0,1)*y+read<real>(t0,2)*z+
+ read<real>(t0,3))*f,
+ (read<real>(t1,0)*x+read<real>(t1,1)*y+read<real>(t1,2)*z+
+ read<real>(t1,3))*f);
+}
+
+// Compute the dot product of vectors a and b.
+real dot(realarray *a, realarray *b)
+{
+ size_t n=checkArrays(a,b);
+ real sum=0.0;
+ for(size_t i=0; i < n; ++i)
+ sum += read<real>(a,i)*read<real>(b,i);
+ return sum;
+}
+
+// Solve the problem L\inv f, where f is an n vector and L is the n x n matrix
+//
+// [ b[0] c[0] a[0] ]
+// [ a[1] b[1] c[1] ]
+// [ a[2] b[2] c[2] ]
+// [ ... ]
+// [ c[n-1] a[n-1] b[n-1] ]
+realarray *tridiagonal(realarray *a, realarray *b, realarray *c, realarray *f)
+{
+ size_t n=checkArrays(a,b);
+ checkEqual(n,checkArray(c));
+ checkEqual(n,checkArray(f));
+
+ array *up=new array(n);
+ array& u=*up;
+
+ if(n == 0) return up;
+
+ // Special case: zero Dirichlet boundary conditions
+ if(read<real>(a,0) == 0.0 && read<real>(c,n-1) == 0.0) {
+ real temp=read<real>(b,0);
+ if(temp == 0.0) dividebyzero();
+ temp=1.0/temp;
+
+ real *work=new real[n];
+ u[0]=read<real>(f,0)*temp;
+ work[0]=-read<real>(c,0)*temp;
+
+ for(size_t i=1; i < n; i++) {
+ real temp=(read<real>(b,i)+read<real>(a,i)*work[i-1]);
+ if(temp == 0.0) {delete[] work; dividebyzero();}
+ temp=1.0/temp;
+ u[i]=(read<real>(f,i)-read<real>(a,i)*read<real>(u,i-1))*temp;
+ work[i]=-read<real>(c,i)*temp;
+ }
+
+ for(size_t i=n-1; i >= 1; i--)
+ u[i-1]=read<real>(u,i-1)+work[i-1]*read<real>(u,i);
+
+ delete[] work;
+ return up;
+ }
+
+ real binv=read<real>(b,0);
+ if(binv == 0.0) dividebyzero();
+ binv=1.0/binv;
+
+ if(n == 1) {u[0]=read<real>(f,0)*binv; return up;}
+ if(n == 2) {
+ real factor=(read<real>(b,0)*read<real>(b,1)-
+ read<real>(a,0)*read<real>(c,1));
+ if(factor== 0.0) dividebyzero();
+ factor=1.0/factor;
+ real temp=(read<real>(b,0)*read<real>(f,1)-
+ read<real>(c,1)*read<real>(f,0))*factor;
+ u[0]=(read<real>(b,1)*read<real>(f,0)-
+ read<real>(a,0)*read<real>(f,1))*factor;
+ u[1]=temp;
+ return up;
+ }
+
+ real *gamma=new real[n-2];
+ real *delta=new real[n-2];
+
+ gamma[0]=read<real>(c,0)*binv;
+ delta[0]=read<real>(a,0)*binv;
+ u[0]=read<real>(f,0)*binv;
+ real beta=read<real>(c,n-1);
+ real fn=read<real>(f,n-1)-beta*read<real>(u,0);
+ real alpha=read<real>(b,n-1)-beta*delta[0];
+
+ for(size_t i=1; i <= n-3; i++) {
+ real alphainv=read<real>(b,i)-read<real>(a,i)*gamma[i-1];
+ if(alphainv == 0.0) {delete[] gamma; delete[] delta; dividebyzero();}
+ alphainv=1.0/alphainv;
+ beta *= -gamma[i-1];
+ gamma[i]=read<real>(c,i)*alphainv;
+ u[i]=(read<real>(f,i)-read<real>(a,i)*read<real>(u,i-1))*alphainv;
+ fn -= beta*read<real>(u,i);
+ delta[i]=-read<real>(a,i)*delta[i-1]*alphainv;
+ alpha -= beta*delta[i];
+ }
+
+ real alphainv=read<real>(b,n-2)-read<real>(a,n-2)*gamma[n-3];
+ if(alphainv == 0.0) {delete[] gamma; delete[] delta; dividebyzero();}
+ alphainv=1.0/alphainv;
+ u[n-2]=(read<real>(f,n-2)-read<real>(a,n-2)*read<real>(u,n-3))
+ *alphainv;
+ beta=read<real>(a,n-1)-beta*gamma[n-3];
+ real dnm1=(read<real>(c,n-2)-read<real>(a,n-2)*delta[n-3])*alphainv;
+ real temp=alpha-beta*dnm1;
+ if(temp == 0.0) {delete[] gamma; delete[] delta; dividebyzero();}
+ u[n-1]=temp=(fn-beta*read<real>(u,n-2))/temp;
+ u[n-2]=read<real>(u,n-2)-dnm1*temp;
+
+ for(size_t i=n-2; i >= 1; i--)
+ u[i-1]=read<real>(u,i-1)-gamma[i-1]*read<real>(u,i)-delta[i-1]*temp;
+
+ delete[] delta;
+ delete[] gamma;
+
+ return up;
+}
+
+// Root solve by Newton-Raphson
+real newton(Int iterations=100, callableReal *f, callableReal *fprime, real x,
+ bool verbose=false)
+{
+ static const real fuzz=1000.0*DBL_EPSILON;
+ Int i=0;
+ size_t oldPrec=0;
+ if(verbose)
+ oldPrec=cout.precision(DBL_DIG);
+
+ real diff=DBL_MAX;
+ real lastdiff;
+ do {
+ real x0=x;
+
+ Stack->push(x);
+ fprime->call(Stack);
+ real dfdx=pop<real>(Stack);
+
+ if(dfdx == 0.0) {
+ x=DBL_MAX;
+ break;
+ }
+
+ Stack->push(x);
+ f->call(Stack);
+ real fx=pop<real>(Stack);
+
+ x -= fx/dfdx;
+
+ lastdiff=diff;
+
+ if(verbose)
+ cout << "Newton-Raphson: " << x << endl;
+
+ diff=fabs(x-x0);
+ if(++i == iterations) {
+ x=DBL_MAX;
+ break;
+ }
+ } while (diff != 0.0 && (diff < lastdiff || diff > fuzz*fabs(x)));
+
+ if(verbose)
+ cout.precision(oldPrec);
+ return x;
+}
+
+// Root solve by Newton-Raphson bisection
+// cf. routine rtsafe (Press et al., Numerical Recipes, 1991).
+real newton(Int iterations=100, callableReal *f, callableReal *fprime, real x1,
+ real x2, bool verbose=false)
+{
+ static const real fuzz=1000.0*DBL_EPSILON;
+ size_t oldPrec=0;
+ if(verbose)
+ oldPrec=cout.precision(DBL_DIG);
+
+ Stack->push(x1);
+ f->call(Stack);
+ real f1=pop<real>(Stack);
+ if(f1 == 0.0) return x1;
+
+ Stack->push(x2);
+ f->call(Stack);
+ real f2=pop<real>(Stack);
+ if(f2 == 0.0) return x2;
+
+ if((f1 > 0.0 && f2 > 0.0) || (f1 < 0.0 && f2 < 0.0)) {
+ ostringstream buf;
+ buf << "root not bracketed, f(x1)=" << f1 << ", f(x2)=" << f2 << endl;
+ error(buf);
+ }
+
+ real x=0.5*(x1+x2);
+ real dxold=fabs(x2-x1);
+ if(f1 > 0.0) {
+ real temp=x1;
+ x1=x2;
+ x2=temp;
+ }
+
+ if(verbose)
+ cout << "midpoint: " << x << endl;
+
+ real dx=dxold;
+ Stack->push(x);
+ f->call(Stack);
+ real y=pop<real>(Stack);
+
+ Stack->push(x);
+ fprime->call(Stack);
+ real dy=pop<real>(Stack);
+
+ Int j;
+ for(j=0; j < iterations; j++) {
+ if(((x-x2)*dy-y)*((x-x1)*dy-y) >= 0.0 || fabs(2.0*y) > fabs(dxold*dy)) {
+ dxold=dx;
+ dx=0.5*(x2-x1);
+ x=x1+dx;
+ if(verbose)
+ cout << "bisection: " << x << endl;
+ if(x1 == x) return x;
+ } else {
+ dxold=dx;
+ dx=y/dy;
+ real temp=x;
+ x -= dx;
+ if(verbose)
+ cout << "Newton-Raphson: " << x << endl;
+ if(temp == x) return x;
+ }
+ if(fabs(dx) < fuzz*fabs(x)) return x;
+
+ Stack->push(x);
+ f->call(Stack);
+ y=pop<real>(Stack);
+
+ Stack->push(x);
+ fprime->call(Stack);
+ dy=pop<real>(Stack);
+
+ if(y < 0.0) x1=x;
+ else x2=x;
+ }
+ if(verbose)
+ cout.precision(oldPrec);
+ return (j == iterations) ? DBL_MAX : x;
+}
+
+real simpson(callableReal *f, real a, real b, real acc=DBL_EPSILON,
+ real dxmax=0)
+{
+ real integral;
+ if(dxmax == 0) dxmax=b-a;
+ Func=f;
+ FuncStack=Stack;
+ if(!simpson(integral,wrapFunction,a,b,acc,dxmax))
+ error("nesting capacity exceeded in simpson");
+ return integral;
+}
+
+// Compute the fast Fourier transform of a pair array
+pairarray* :pairArrayFFT(pairarray *a, Int sign=1)
+{
+ unsigned n=(unsigned) checkArray(a);
+#ifdef HAVE_LIBFFTW3
+ array *c=new array(n);
+ if(n) {
+ Complex *f=FFTWComplex(n);
+ fft1d Forward(n,intcast(sign),f);
+
+ for(size_t i=0; i < n; i++) {
+ pair z=read<pair>(a,i);
+ f[i]=Complex(z.getx(),z.gety());
+ }
+ Forward.fft(f);
+
+ for(size_t i=0; i < n; i++) {
+ Complex z=f[i];
+ (*c)[i]=pair(z.real(),z.imag());
+ }
+ FFTWdelete(f);
+ }
+#else
+ unused(&n);
+ unused(&sign);
+ array *c=new array(0);
+#endif // HAVE_LIBFFTW3
+ return c;
+}
+
+Intarray2 *triangulate(pairarray *z)
+{
+ size_t nv=checkArray(z);
+// Call robust version of Gilles Dumoulin's port of Paul Bourke's
+// triangulation code.
+
+ XYZ *pxyz=new XYZ[nv+3];
+ ITRIANGLE *V=new ITRIANGLE[4*nv];
+
+ for(size_t i=0; i < nv; ++i) {
+ pair w=read<pair>(z,i);
+ pxyz[i].p[0]=w.getx();
+ pxyz[i].p[1]=w.gety();
+ pxyz[i].i=(Int) i;
+ }
+
+ Int ntri;
+ Triangulate((Int) nv,pxyz,V,ntri,true,false);
+
+ size_t nt=(size_t) ntri;
+ array *t=new array(nt);
+ for(size_t i=0; i < nt; ++i) {
+ array *ti=new array(3);
+ (*t)[i]=ti;
+ ITRIANGLE *Vi=V+i;
+ (*ti)[0]=pxyz[Vi->p1].i;
+ (*ti)[1]=pxyz[Vi->p2].i;
+ (*ti)[2]=pxyz[Vi->p3].i;
+ }
+
+ delete[] V;
+ delete[] pxyz;
+ return t;
+}
+
+real norm(realarray *a)
+{
+ size_t n=checkArray(a);
+ real M=0.0;
+ for(size_t i=0; i < n; ++i) {
+ real x=fabs(vm::read<real>(a,i));
+ if(x > M) M=x;
+ }
+ return M;
+}
+
+real norm(realarray2 *a)
+{
+ size_t n=checkArray(a);
+ real M=0.0;
+ for(size_t i=0; i < n; ++i) {
+ vm::array *ai=vm::read<vm::array*>(a,i);
+ size_t m=checkArray(ai);
+ for(size_t j=0; j < m; ++j) {
+ real a=fabs(vm::read<real>(ai,j));
+ if(a > M) M=a;
+ }
+ }
+ return M;
+}
+
+real norm(triplearray2 *a)
+{
+ size_t n=checkArray(a);
+ real M=0.0;
+ for(size_t i=0; i < n; ++i) {
+ vm::array *ai=vm::read<vm::array*>(a,i);
+ size_t m=checkArray(ai);
+ for(size_t j=0; j < m; ++j) {
+ real a=vm::read<triple>(ai,j).abs2();
+ if(a > M) M=a;
+ }
+ }
+ return sqrt(M);
+}
+
+real change2(triplearray2 *a)
+{
+ size_t n=checkArray(a);
+ if(n == 0) return 0.0;
+
+ vm::array *a0=vm::read<vm::array*>(a,0);
+ size_t m=checkArray(a0);
+ if(m == 0) return 0.0;
+ triple a00=vm::read<triple>(a0,0);
+ real M=0.0;
+
+ for(size_t i=0; i < n; ++i) {
+ vm::array *ai=vm::read<vm::array*>(a,i);
+ size_t m=checkArray(ai);
+ for(size_t j=0; j < m; ++j) {
+ real a=(vm::read<triple>(ai,j)-a00).abs2();
+ if(a > M) M=a;
+ }
+ }
+ return M;
+}
+
+triple minbezier(triplearray2 *P, triple b)
+{
+ real *A=copyTripleArray2Components(P,true,4);
+ b=triple(bound(A,::min,b.getx(),sqrtFuzz*norm(A,16)),
+ bound(A+16,::min,b.gety(),sqrtFuzz*norm(A+16,16)),
+ bound(A+32,::min,b.getz(),sqrtFuzz*norm(A+32,16)));
+ delete[] A;
+ return b;
+}
+
+triple maxbezier(triplearray2 *P, triple b)
+{
+ real *A=copyTripleArray2Components(P,true,4);
+ b=triple(bound(A,::max,b.getx(),sqrtFuzz*norm(A,16)),
+ bound(A+16,::max,b.gety(),sqrtFuzz*norm(A+16,16)),
+ bound(A+32,::max,b.getz(),sqrtFuzz*norm(A+32,16)));
+ delete[] A;
+ return b;
+}
+
+pair minratio(triplearray2 *P, pair b)
+{
+ triple *A=copyTripleArray2C(P,true,4);
+ real fuzz=sqrtFuzz*norm(A,16);
+ b=pair(bound(A,::min,xratio,b.getx(),fuzz),
+ bound(A,::min,yratio,b.gety(),fuzz));
+ delete[] A;
+ return b;
+}
+
+pair maxratio(triplearray2 *P, pair b)
+{
+ triple *A=copyTripleArray2C(P,true,4);
+ real fuzz=sqrtFuzz*norm(A,16);
+ b=pair(bound(A,::max,xratio,b.getx(),fuzz),
+ bound(A,::max,yratio,b.gety(),fuzz));
+ delete[] A;
+ return b;
+}
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