SpectralFuncs.hpp

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#ifndef SPECTRALFUNCS_HPP
#define SPECTRALFUNCS_HPP

#include <cfloat>

//======================================================
// from types.h
//======================================================

/* integer type to use for everything */
#if defined( USE_LONG )
#define INTEGER long
#elif defined( USE_LONG_LONG )
#define INTEGER long long
#elif defined( USE_SHORT )
#define INTEGER short
#else
#define INTEGER int
#endif

/* when defined, use the given type for global indices instead of INTEGER */
#if defined( USE_GLOBAL_LONG_LONG )
#define GLOBAL_INT long long
#elif defined( USE_GLOBAL_LONG )
#define GLOBAL_INT long
#else
#define GLOBAL_INT long
#endif

/* floating point type to use for everything */
#if defined( USE_FLOAT )
typedef float real;
#define floorr floorf
#define ceilr  ceilf
#define sqrtr  sqrtf
#define fabsr  fabsf
#define cosr   cosf
#define sinr   sinf
#define EPS    ( 128 * FLT_EPSILON )
#define PI     3.1415926535897932384626433832795028841971693993751058209749445923F
#elif defined( USE_LONG_DOUBLE )
typedef long double real;
#define floorr floorl
#define ceilr  ceill
#define sqrtr  sqrtl
#define fabsr  fabsl
#define cosr   cosl
#define sinr   sinl
#define EPS    ( 128 * LDBL_EPSILON )
#define PI     3.1415926535897932384626433832795028841971693993751058209749445923L
#else
typedef double real;
#define floorr floor
#define ceilr  ceil
#define sqrtr  sqrt
#define fabsr  fabs
#define cosr   cos
#define sinr   sin
#define EPS    ( 128 * DBL_EPSILON )
#define PI     3.1415926535897932384626433832795028841971693993751058209749445923
#endif

/* apparently uint and ulong can be defined already in standard headers */
#define uint  uint_
#define ulong ulong_
#define sint  sint_
#define slong slong_

typedef signed INTEGER sint;
typedef unsigned INTEGER uint;
#undef INTEGER

#ifdef GLOBAL_INT
typedef signed GLOBAL_INT slong;
typedef unsigned GLOBAL_INT ulong;
#else
typedef sint slong;
typedef uint ulong;
#endif

//======================================================
// from poly.h
//======================================================

/*
  For brevity's sake, some names have been shortened
  Quadrature rules
    Gauss   -> Gauss-Legendre quadrature (open)
    Lobatto -> Gauss-Lobatto-Legendre quadrature (closed at both ends)
  Polynomial bases
    Legendre -> Legendre basis
    Gauss    -> Lagrangian basis using Gauss   quadrature nodes
    Lobatto  -> Lagrangian basis using Lobatto quadrature nodes
*/

/*--------------------------------------------------------------------------
   Legendre Polynomial Matrix Computation
   (compute P_i(x_j) for i = 0, ..., n and a given set of x)
  --------------------------------------------------------------------------*/

/* precondition: n >= 1
   inner index is x index (0 ... m-1);
   outer index is Legendre polynomial number (0 ... n)
 */
void legendre_matrix( const real* x, int m, real* P, int n );

/* precondition: n >= 1
   inner index is Legendre polynomial number (0 ... n)
   outer index is x index (0 ... m-1);
 */
void legendre_matrix_t( const real* x, int m, real* P, int n );

/* precondition: n >= 1
   compute P_i(x) with i = 0 ... n
 */
void legendre_row( real x, real* P, int n );

/*--------------------------------------------------------------------------
   Quadrature Nodes and Weights Calculation

   call the _nodes function before calling the _weights function
  --------------------------------------------------------------------------*/

void gauss_nodes( real* z, int n );   /* n nodes (order = 2n-1) */
void lobatto_nodes( real* z, int n ); /* n nodes (order = 2n-3) */

void gauss_weights( const real* z, real* w, int n );
void lobatto_weights( const real* z, real* w, int n );

/*--------------------------------------------------------------------------
   Lagrangian to Legendre Change-of-basis Matrix
  --------------------------------------------------------------------------*/

/* precondition: n >= 2
   given the Gauss quadrature rule (z,w,n), compute the square matrix J
   for transforming from the Gauss basis to the Legendre basis:

      u_legendre(i) = sum_j J(i,j) u_gauss(j)

   computes J   = .5 (2i+1) w  P (z )
             ij              j  i  j

   in column major format (inner index is i, the Legendre index)
 */
void gauss_to_legendre( const real* z, const real* w, int n, real* J );

/* precondition: n >= 2
   same as above, but
   in row major format (inner index is j, the Gauss index)
 */
void gauss_to_legendre_t( const real* z, const real* w, int n, real* J );

/* precondition: n >= 3
   given the Lobatto quadrature rule (z,w,n), compute the square matrix J
   for transforming from the Lobatto basis to the Legendre basis:

      u_legendre(i) = sum_j J(i,j) u_lobatto(j)

   in column major format (inner index is i, the Legendre index)
 */
void lobatto_to_legendre( const real* z, const real* w, int n, real* J );

/*--------------------------------------------------------------------------
   Lagrangian basis function evaluation
  --------------------------------------------------------------------------*/

/* given the Lagrangian nodes (z,n) and evaluation points (x,m)
   evaluate all Lagrangian basis functions at all points x

   inner index of output J is the basis function index (row-major format)
   provide work array with space for 4*n doubles
 */
void lagrange_weights( const real* z, unsigned n, const real* x, unsigned m, real* J, real* work );

/* given the Lagrangian nodes (z,n) and evaluation points (x,m)
   evaluate all Lagrangian basis functions and their derivatives

   inner index of outputs J,D is the basis function index (row-major format)
   provide work array with space for 6*n doubles
 */
void lagrange_weights_deriv( const real* z, unsigned n, const real* x, unsigned m, real* J, real* D, real* work );

/*--------------------------------------------------------------------------
   Speedy Lagrangian Interpolation

   Usage:

     lagrange_data p;
     lagrange_setup(&p,z,n);    *  setup for nodes z[0 ... n-1] *

     the weights
       p->J [0 ... n-1]     interpolation weights
       p->D [0 ... n-1]     1st derivative weights
       p->D2[0 ... n-1]     2nd derivative weights
     are computed for a given x with:
       lagrange_0(p,x);  *  compute p->J *
       lagrange_1(p,x);  *  compute p->J, p->D *
       lagrange_2(p,x);  *  compute p->J, p->D, p->D2 *
       lagrange_2u(p);   *  compute p->D2 after call of lagrange_1(p,x); *
     These functions use the z array supplied to setup
       (that pointer should not be freed between calls)
     Weights for x=z[0] and x=z[n-1] are computed during setup; access as:
       p->J_z0, etc. and p->J_zn, etc.

     lagrange_free(&p);  *  deallocate memory allocated by setup *
  --------------------------------------------------------------------------*/

typedef struct
{
    unsigned n;                                /* number of Lagrange nodes            */
    const real* z;                             /* Lagrange nodes (user-supplied)      */
    real *J, *D, *D2;                          /* weights for 0th,1st,2nd derivatives */
    real *J_z0, *D_z0, *D2_z0;                 /* ditto at z[0]   (computed at setup) */
    real *J_zn, *D_zn, *D2_zn;                 /* ditto at z[n-1] (computed at setup) */
    real *w, *d, *u0, *v0, *u1, *v1, *u2, *v2; /* work data           */
} lagrange_data;

void lagrange_setup( lagrange_data* p, const real* z, unsigned n );
void lagrange_free( lagrange_data* p );

void lagrange_0( lagrange_data* p, real x );
void lagrange_1( lagrange_data* p, real x );
void lagrange_2( lagrange_data* p, real x );
void lagrange_2u( lagrange_data* p );

//======================================================
// from tensor.h
//======================================================

/*--------------------------------------------------------------------------
   1-,2-,3-d Tensor Application

   the 3d case:
   tensor_f3(R,mr,nr, S,ms,ns, T,mt,nt, u,v, work1,work2)
     gives v = [ R (x) S (x) T ] u
     where R is mr x nr, S is ms x ns, T is mt x nt,
       each in row- or column-major format according to f := r | c
     u is nr x ns x nt in column-major format (inner index is r)
     v is mr x ms x mt in column-major format (inner index is r)
  --------------------------------------------------------------------------*/

void tensor_c1( const real* R, unsigned mr, unsigned nr, const real* u, real* v );
void tensor_r1( const real* R, unsigned mr, unsigned nr, const real* u, real* v );

/* work holds mr*ns reals */
void tensor_c2( const real* R,
                unsigned mr,
                unsigned nr,
                const real* S,
                unsigned ms,
                unsigned ns,
                const real* u,
                real* v,
                real* work );
void tensor_r2( const real* R,
                unsigned mr,
                unsigned nr,
                const real* S,
                unsigned ms,
                unsigned ns,
                const real* u,
                real* v,
                real* work );

/* work1 holds mr*ns*nt reals,
   work2 holds mr*ms*nt reals */
void tensor_c3( const real* R,
                unsigned mr,
                unsigned nr,
                const real* S,
                unsigned ms,
                unsigned ns,
                const real* T,
                unsigned mt,
                unsigned nt,
                const real* u,
                real* v,
                real* work1,
                real* work2 );
void tensor_r3( const real* R,
                unsigned mr,
                unsigned nr,
                const real* S,
                unsigned ms,
                unsigned ns,
                const real* T,
                unsigned mt,
                unsigned nt,
                const real* u,
                real* v,
                real* work1,
                real* work2 );

/*--------------------------------------------------------------------------
   1-,2-,3-d Tensor Application of Row Vectors (for Interpolation)

   the 3d case:
   v = tensor_i3(Jr,nr, Js,ns, Jt,nt, u, work)
   same effect as tensor_r3(Jr,1,nr, Js,1,ns, Jt,1,nt, u,&v, work1,work2):
     gives v = [ Jr (x) Js (x) Jt ] u
     where Jr, Js, Jt are row vectors (interpolation weights)
     u is nr x ns x nt in column-major format (inner index is r)
     v is a scalar
  --------------------------------------------------------------------------*/

real tensor_i1( const real* Jr, unsigned nr, const real* u );

/* work holds ns reals */
real tensor_i2( const real* Jr, unsigned nr, const real* Js, unsigned ns, const real* u, real* work );

/* work holds ns*nt + nt reals */
real tensor_i3( const real* Jr,
                unsigned nr,
                const real* Js,
                unsigned ns,
                const real* Jt,
                unsigned nt,
                const real* u,
                real* work );

/*--------------------------------------------------------------------------
   1-,2-,3-d Tensor Application of Row Vectors
             for simultaneous Interpolation and Gradient computation

   the 3d case:
   v = tensor_ig3(Jr,Dr,nr, Js,Ds,ns, Jt,Dt,nt, u,g, work)
     gives v   = [ Jr (x) Js (x) Jt ] u
           g_0 = [ Dr (x) Js (x) Jt ] u
           g_1 = [ Jr (x) Ds (x) Jt ] u
           g_2 = [ Jr (x) Js (x) Dt ] u
     where Jr,Dr,Js,Ds,Jt,Dt are row vectors
       (interpolation & derivative weights)
     u is nr x ns x nt in column-major format (inner index is r)
     v is a scalar, g is an array of 3 reals
  --------------------------------------------------------------------------*/

real tensor_ig1( const real* Jr, const real* Dr, unsigned nr, const real* u, real* g );

/* work holds 2*ns reals */
real tensor_ig2( const real* Jr,
                 const real* Dr,
                 unsigned nr,
                 const real* Js,
                 const real* Ds,
                 unsigned ns,
                 const real* u,
                 real* g,
                 real* work );

/* work holds 2*ns*nt + 3*ns reals */
real tensor_ig3( const real* Jr,
                 const real* Dr,
                 unsigned nr,
                 const real* Js,
                 const real* Ds,
                 unsigned ns,
                 const real* Jt,
                 const real* Dt,
                 unsigned nt,
                 const real* u,
                 real* g,
                 real* work );

//======================================================
// from findpt.h
//======================================================

typedef struct
{
    const real* xw[2];               /* geometry data */
    real* z[2];                      /* lobatto nodes */
    lagrange_data ld[2];             /* interpolation, derivative weights & data */
    unsigned nptel;                  /* nodes per element */
    struct findpt_hash_data_2* hash; /* geometric hashing data */
    struct findpt_listel *list, **sorted, **end;
    /* pre-allocated list of elements to
       check (found by hashing), and
       pre-allocated list of pointers into
       the first list for sorting */
    struct findpt_opt_data_2* od; /* data for the optimization algorithm */
    real* od_work;
} findpt_data_2;

typedef struct
{
    const real* xw[3];               /* geometry data */
    real* z[3];                      /* lobatto nodes */
    lagrange_data ld[3];             /* interpolation, derivative weights & data */
    unsigned nptel;                  /* nodes per element */
    struct findpt_hash_data_3* hash; /* geometric hashing data */
    struct findpt_listel *list, **sorted, **end;
    /* pre-allocated list of elements to
       check (found by hashing), and
       pre-allocated list of pointers into
       the first list for sorting */
    struct findpt_opt_data_3* od; /* data for the optimization algorithm */
    real* od_work;
} findpt_data_3;

findpt_data_2* findpt_setup_2( const real* const xw[2],
                               const unsigned n[2],
                               uint nel,
                               uint max_hash_size,
                               real bbox_tol );
findpt_data_3* findpt_setup_3( const real* const xw[3],
                               const unsigned n[3],
                               uint nel,
                               uint max_hash_size,
                               real bbox_tol );

void findpt_free_2( findpt_data_2* p );
void findpt_free_3( findpt_data_3* p );

const real* findpt_allbnd_2( const findpt_data_2* p );
const real* findpt_allbnd_3( const findpt_data_3* p );

typedef int ( *findpt_func )( void*, const real*, int, uint*, real*, real* );
int findpt_2( findpt_data_2* p, const real x[2], int guess, uint* el, real r[2], real* dist );
int findpt_3( findpt_data_3* p, const real x[3], int guess, uint* el, real r[3], real* dist );

inline void findpt_weights_2( findpt_data_2* p, const real r[2] )
{
    lagrange_0( &p->ld[0], r[0] );
    lagrange_0( &p->ld[1], r[1] );
}

inline void findpt_weights_3( findpt_data_3* p, const real r[3] )
{
    lagrange_0( &p->ld[0], r[0] );
    lagrange_0( &p->ld[1], r[1] );
    lagrange_0( &p->ld[2], r[2] );
}

inline double findpt_eval_2( findpt_data_2* p, const real* u )
{
    return tensor_i2( p->ld[0].J, p->ld[0].n, p->ld[1].J, p->ld[1].n, u, p->od_work );
}

inline double findpt_eval_3( findpt_data_3* p, const real* u )
{
    return tensor_i3( p->ld[0].J, p->ld[0].n, p->ld[1].J, p->ld[1].n, p->ld[2].J, p->ld[2].n, u, p->od_work );
}

//======================================================
// from extrafindpt.h
//======================================================

typedef struct
{
    unsigned constraints;
    unsigned dn, d1, d2;
    real *x[3], *fdn[3];
} opt_face_data_3;

typedef struct
{
    unsigned constraints;
    unsigned de, d1, d2;
    real *x[3], *fd1[3], *fd2[3];
} opt_edge_data_3;

typedef struct
{
    unsigned constraints;
    real x[3], jac[9];
} opt_point_data_3;

typedef struct
{
    lagrange_data* ld;
    unsigned size[4];
    const real* elx[3];
    opt_face_data_3 fd;
    opt_edge_data_3 ed;
    opt_point_data_3 pd;
    real* work;
    real x[3], jac[9];
} opt_data_3;

void opt_alloc_3( opt_data_3* p, lagrange_data* ld );
void opt_free_3( opt_data_3* p );
double opt_findpt_3( opt_data_3* p, const real* const elx[3], const real xstar[3], real r[3], unsigned* constr );
void opt_vol_set_intp_3( opt_data_3* p, const real r[3] );

const unsigned opt_no_constraints_2 = 3 + 1;
const unsigned opt_no_constraints_3 = 9 + 3 + 1;

/* for 2d spectralQuad */
/*--------------------------------------------------------------------------

   2 - D

  --------------------------------------------------------------------------*/

typedef struct
{
    unsigned constraints;
    unsigned de, d1;
    real *x[2], *fd1[2];
} opt_edge_data_2;

typedef struct
{
    unsigned constraints;
    real x[2], jac[4];
} opt_point_data_2;

typedef struct
{
    lagrange_data* ld;
    unsigned size[3];
    const real* elx[2];
    opt_edge_data_2 ed;
    opt_point_data_2 pd;
    real* work;
    real x[2], jac[4];
} opt_data_2;
void opt_alloc_2( opt_data_2* p, lagrange_data* ld );
void opt_free_2( opt_data_2* p );
double opt_findpt_2( opt_data_2* p, const real* const elx[2], const real xstar[2], real r[2], unsigned* constr );

#endif