Construction of the SG²
Function for SG
Functions for SG²
Operations of Functions on the Gasket
Functions for the Finite Element Method
Functions Specific to Certain Equations
I/O Functions
 

1  Construction of the SG²

The SG² is simply the product of two Sierpinski gaskets. Thus, many of the functions combine two factors of SG.

1.1  pd_gasket

Description: Makes a gasket of regular decomposition. Function: pd_gasket(m)
Arguments: m - the depth of the regular decomposition of the SG²
Output: Gasket of depth m.

1.2  pd_gasket_irreg

Description: Makes a gasket of irregular decomposition.
Function: pd_gasket_irreg(wordlist)
Arguments: wordlist - a product wordlist of cells to extend the gasket to. Usually, one makes an irregular decomposition with Kevin Coletta's reg_wordlist, zoom_wordlist, and subdivid_irreg, then calls the wordlist_makeprod to get a product wordlist
Output: Irregularly decomposed gasket, as described by wordlist
 

1.3  wordlist_makeprod

Description: Calculates the product of a wordlist with itself (that describes a decomposition of SG) for SG²
Function: wordlist_makeprod(wordlist)
Arguments: wordlist - decomposition of an SG
Output: Product of two wordlist's.
 

1.4  wordlist_combine

Description: Calculates the product of two wordlists for SG²
Function: wordlist_combine(wordlist1,wordlist2)
Arguments: wordlist - decomposition of SGў; wordlist2 - decomposition of SGўў.
Output: Product wordlist of the two wordlist arguments.

1.5  pd_vertices_irreg

Description: Computes a list of vertices
Function: pd_vertices_irreg(wordlist)
Arguments: wordlist - decomposition of an SG²
Output: List of vertices, decomposed according to wordlist
 

1.6  pd_vertices

Description: Computes a list of vertices, regularly decomposed. WARNING - may be slow for detailed SG²
Function: pd_vertices(m)
Arguments: m - depth of vertices
Output: List of vertices of SG², decomposed to level m.
 

1.7  pd_calc_biharms

Description: Generates the pluriharmonic basis elements for SG². There are 27 basis elements in all. Look through the paper for a description of them.
Function: pd_calc_biharms(T,R,vm,SG_lvl,func)
Arguments: T - product gasket. R - throwback to Gibbons code; it helps create biharmonic functions on the SG. Always pass R. vm - list of vertices for T. SG_lvl - depth of the Sierpinski gasket to compute to. func - function number to save the first basis element to. pd_calc_biharms will utilize every function slot from func to func+27.
Output: The 27 pluribiharmonic basis functions will be saved in T (to depth SG_lvl) in function slots func to func+27.
 

1.8  wordlist_seperate

Description: Seperates the product wordlist in the argument to two wordlists in the factors (properly sorted!).
Function: wordlist_seperate(wordlist)
Arguments: wordlist - a product wordlist to seperate.
Output: The return value is a list of the two seperate wordlists, ie, [retVal1,retVal2], where retVal1 is the wordlist for the first factor and retVal2 is the wordlist for the second factor.
 

1.9  pd_make_neighbors

Description: Determines the neighboring points of a SG², and saves them to the gasket
Function: pd_make_neighbors(T,m_basis,vert)
Arguments: T - gasket to save the neighbors to (regular decomposition). m_basis - depth to compute vertices to. vert - vertex list of T.
Output: Function slots 1 through 16 in T are filled up with the neighbors of points in T.
 

1.10  pd_make_neighbors_irreg

Description: Determines the neighboring points of a SG², and saves them to the gasket
Function: pd_make_neighbors_irreg(T,W)
Arguments: T - gasket to save the neighbors to (irregular decomposition). W - decomposition of T.
Output: Function slots 1 through 16 in T are filled up with the neighbors of points in T.

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2  Functions for SG

2.1  isElementOf

Description: Determines if a point is an element of a gasket (works with SG² also)
Function: isElementOf(T,elem)
Arguments: T - gasket to analyze. elem - possible point within that gasket.
Output: The function returns true if elem is within T, false otherwise.
 

2.2  calc_cell_address

Description: Calculates the cell address on the SG of a given 3 points.
Function: calc_cell_address(points)
Arguments: points - list of 3 points which the procedure will calculate the cell address of.

2.3  next_cells

Description: Calculates the 3 immediate subcells of a given cell.
Function: next_cells(cell)
Arguments: cell - SG cell to ``break-apart".
Output: The 3 subcells of cell.

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3  Functions for SG²

3.1  pd_Cross

Description: Makes the cross product of its arguments
Function: pd_Cross(cell1, cell2)
Arguments: cell1 - cell of SGў. cell2 - cell of SGўў.
Output: The cross product of cell1 and cell2, a cell in SG²

3.2  bottom_lvl

Description: Determines if a given cell is at the bottom level of an SG².
Function: bottom_lvl(T,cell)
Arguments: T - gasket to analyze. cell - cell (list of 9 points) to work with.
Output: The function returns true if cell is in the bottommost level of T, false otherwise.

3.3  pd_points_lower

Description: Finds all the points within a cell.
Function: pd_points_lower(T, cell)
Arguments: T - gasket we are working with. cell - cell (list of 9 points) to find the points in.
Output: The function returns a list of points that are within cell, including the boundary of cell.

3.4  pd_cross_func

Description: Creates a function on SG² as the product of two SG functions.
Function: pd_cross_func(T, vm, SG_T, SG_T2, func1, func2, work)
Arguments: T - gasket to save the new function to. vm - vertex list that describes T. SG_T - SG used as factor 1. SG_T2 - SG used as factor 2. func1 - function within SG_T to use. func2 - function within SG_T2 to use. work - function in T to save the cross to.
Output: The function func1 multiplied by func2, saved in T in function slot work.

3.5  pd_next_cells

Description: Calculates the 9 immediate subcells of a given cell.
Function: pd_next_cells(cell)
Arguments: cell - SG² cell to ``break-apart".
Output: The 9 subcells of cell.

3.6  pd_calc_cell_address

Description: Calculates the cell address of a given 9 points.
Function: pd_calc_cell_address(points)
Arguments: points - list of 9 points which the procedure will calculate the cell address of.

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4  Operations of Functions on the Gasket

4.1  pd_integrate_func

Description: Integrates a function.
Function: pd_integrate_func(T,func_pl,{Cell})
Arguments: T - a gasket with a function described in func_pl. func_pl - the slot of the function to be integrated. Cell - this is an optional argument. If you just want to integrate a function inside a certain cell (including the boundary), specify the 9 boundary points of that cell here. Must be a valid cell (There is no error checking to see if this is valid. I have no clue what will happen if it isn't one - ideally it'd cause an error, but it may just give you an invalid answer. If the Cell argument is not given, then the integral is taken over the entire gasket.
Output: The integral of the function func_pl in T on Cell.
 

4.2  pd_integrate_func_basis

Description: Computes тf fdm for a general function f and a basis element f. f is selected by argument i1 - the point where f = 1 if it is of f type, or where ¶_n f = 1 if f is of g type.
Function: pd_integrate_func_basis(T,S,func_pl,harmonic_pl,work_pl, i1)
Arguments: T - gasket that includes f. S - spline gasket. func_pl - slot in T where f is stored. harmonic_pl - slot in T where the first PBH basis element is stored. work_pl - empty slot where temporary work can be placed. i1 - point which signifies basis element.
Output: The integral of the function times the basis element in T.

4.3  pd_elem_mult

Description: Multiplies two functions together.
Function: pd_elem_mult(T,func1,func2,work,{Cell})
Arguments: T - gasket that contains the two functions to multiply. func1 - first function to multiply. func2 - other function. work - where to store the output of the function. {Cell} - optional argument, the cell to multiply the functions over.
Output: The two input functions multiplied together and stored in work. If Cell is given, the function only works over that cell. func2 is resized to fit into that cell, and func1 is multiplied against the resized func2.
 

4.4  pd_Laplacian

Description: Calculates the Laplacian of a function.
Function: pd_Laplacian(T,func,work)
Arguments: T - gasket that is used. func - function to take the Laplacian of. work - function slot to replace result in.
Output: Stores the Laplacian of func in work in T.
 

4.5  set_const

Description: Sets a function slot to a constant.
Function: set_const(T, vm, func, const)
Arguments: T - gasket to work with. vm - corresponding vertex list. func - function to set constant. const - constant to set function to.
Output: Function in func in T is identically const.

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5  Functions for the Finite Element Method

5.1  pd_calc_biharms

Description: Calculates the biharmonic basis functions for the product, and saves them to a gasket.
Function: pd_calc_biharms(T,R,vm,SG_lvl,func)
Arguments: T - gasket to save basis functions to. R - always pass the second argument as R (throwback to the older Gibbons code). vm - vertex list that corresponds to T. SG_lvl - the deepest level in the gasket to compute to (because the underlying code is not set up for irregular decomposition). func - function to save the biharmonic basis to. There are 27 functions, so this procedure will use up each slot from func to func+27.
Output: The biharmonic basis functions in slots func to func+27 in T.

5.2  pd_reconstruct_func

Description: Reconstructs a function to point values, given a vector that expresses the function in terms of the spline basis.
Function: pd_reconstruct_func(T,W,vm2,sol,harm,func)
Arguments: T - gasket to reconstruct the function to, also holds the biharmonic basis. W - wordlist that decomposes the SPLINE (not the gasket T). vm2 - vertex list for the GASKET (not the spline). sol - valid solution vector. harm - number corresponding to the function slot beginning of the harmonic basis functions in T. func - function slot to save the reconstructed function to.
Output: The reconstructed function in slot func in T.

5.3  pd_integrate_f_basis

Description: Integrates a function against all of the spline basis elements, returns a vector, which is the function representation in the spline space.
Function: pd_integrate_f_basis(T,S,W2,W,func,harm,work)
Arguments: T - gasket containing the function. S - spline to use. W2 - wordlist that corresponds to T. W - wordlist corresponding to S. func - function to integrate against. harm - start of the harmonic functions. work - slot that the procedure can save its temporary work to (any previous data will be destroyed)
Output: A vector that describes the function in the spline basis.
 

5.4  pd_func_vector

Description: A vector that describes the order of the spline basis functions.
Function: pd_func_vector(W)
Arguments: W - wordlist that describes the spline decomposition.
Output: A vector that describes the spline basis functions. See the internal data structures section for more info.

5.5  pd_gen_energy_matrix_h

Description: Generates the energy matrix for harmonic functions, based on regular decomposition.
Function: pd_gen_energy_matrix_h(S,m,vm)
Arguments: S - the spline gasket. m - approximation depth of gasket depth to create the energy matrix for. vm - vertex list corresponding to S.
Output: A vm-by-vm matrix that describes the energy of the pluriharmonic basis elements.
 

5.6  pd_gen_energy_matrix

Description: Generates the energy matrix for biharmonic functions, based on regular decomposition.
Function: pd_gen_energy_matrix_h(S,m,vm)
Arguments: S - the spline gasket. m - approximation depth of gasket to create the energy matrix for. vm - vertex list corresponding to S.
Output: A matrix that describes the energy of the pluribiharmonic basis elements.
 

5.7  pd_gen_energy_matrix_irreg

Description: Generates the energy matrix for biharmonic functions, based on irregular decomposition.
Function: pd_gen_energy_matrix_h(S,W)
Arguments: S - the spline gasket. W - wordlist decomposition of S
Output: A matrix that describes the energy of the pluribiharmonic basis elements.
 

5.8  pd_gen_energy_matrix_irreg_2

Description: Generates the energy matrix for biharmonic functions, based on irregular decomposition and an alternate differential operator.
Function: pd_gen_energy_matrix_h(S,W,val)
Arguments: S - the spline gasket. W - wordlist decomposition of S. val - constant for the differential operator. Reference the ``Other Differential Operators" section.
Output: A matrix that describes the energy of the pluribiharmonic basis elements, based on the differential operator D = Dў- val Dўў

5.9  pd_gen_iprod_matrix_h

Description: Generates the Gram matrix for harmonic functions, based on regular decomposition.
Function: pd_gen_iprod_matrix_h(S,m,vm)
Arguments: S - the spline gasket. m - approximation depth of gasket depth to create the energy matrix for. vm - vertex list corresponding to S.
Output: A vm-by-vm matrix that describes the integral of the pluriharmonic basis elements.
 

5.10  pd_gen_iprod_matrix

Description: Generates the Gram matrix for biharmonic functions, based on regular decomposition.
Function: pd_gen_iprod_matrix_h(S,m,vm)
Arguments: S - the spline gasket. m - approximation depth of gasket to create the energy matrix for. vm - vertex list corresponding to S.
Output: A matrix that describes the integral of the pluribiharmonic basis elements.
 

5.11  pd_gen_iprod_matrix_irreg

Description: Generates the Gram matrix for biharmonic functions, based on irregular decomposition.
Function: pd_gen_iprod_matrix_h(S,W)
Arguments: S - the spline gasket. W - wordlist decomposition of S
Output: A matrix that describes the integral of the pluribiharmonic basis elements.

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6  I/O Functions

Section to come!

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7  Functions specific to certain equations

Still working on this section!