13. Range Functions
Range functions take an argument list of expressions and ranges and return one or more values.The range functions are:
avg average of the defined cells
count number of cells defined
dot dot product (inner product) of two ranges
feval val=feval("type",cr,x), evaluate parameterized function
find {n,cr}=find(xr,value), find value in range
llsq {rank,cr}=llsq("type",xr,yr), linear least squares
majority non-zero if majority of defined cells are non-zero
max maximum of the defined cells
min minimum of the defined cells
prod product of the defined cells
search {xr}=search(f,x0[,dx[,x1...xN]]), search for minimum of f(xr)
stats {a,s,l,h}=stats(...), avg, stdev, min, and max of the defined cells
stdev standard deviation of the defined cells
sum sum of the defined cells
var variance of the defined cells
Additional information:
dot dot product (inner product) of two ranges
- The dot product dot(x,y) is the sum of the pairwise products of the elements
of the x and y ranges, which must have the same size:
dot(x,y) = x0 y0 + x1 y1 + x2 y2 + ...
feval val=feval("type",cr,x), evaluate parameterized function- The function is evaluated at the point x using parameters specified
by the range cr.
Available function types are:
- "poly"
- A polynomial in the following form:
c0 + c1 x + c2 x2 + c3 x3 + ...
The polynomial is evaluated using Horner's rule. For example, using 4 coefficients:c0 + x*(c1 + x*(c2 + x*c3))
- (When other function types are implemented they will be documented here)
find {n,cr}=find(xr,value), find value in range- Finds column and row positions which contain the specified value.
Searches the defined cells in the range xr and produces a count n of how many cells match the specified value. The column and row positions of the matching cells are stored in the cr result range, if it is specified. The extent of the range filled is determined by the number of matches rather than by the size of the range. The end cell of the range determines only the direction for traversing the range. This is similar to the fill...{ expr_list } command.
If cr is specified, it must consist of two rows or two columns, in order to store the column and row pair values in adjacent cells. If cr has two rows and two columns, the default direction (byrows or bycols) is used to store the results; otherwise, if cr has two rows the result pairs are stored by columns and if it has two columns they are stored by rows.
Example:
% cat find.ss
d1:e3 = { 66, 99, 66, 66, 99, 77 }; {a0,a1:b2} = find( d1:e3, 99); {a4,a5:a6} = find( d1:e3, 66); eval; format "%g"; format CR; print;% SS find.ss 0 1 2 3 4 0 2 1 4 1 66 99 2 3 3 66 66 3 99 77 4 3 5 3 3 4 6 1 2 2
llsq {rank,cr}=llsq("type",xr,yr), linear least squares- Solves for the least square error approximation for y as a function of
x.
If type is "data", then no basis functions are used, and y is approximated as a direct linear combination of the data with coefficients specified by the range cr:
y = c0 x0 + c1 x1 + c2 x2 + ...
The xr input data range size must equal the product of the sizes of cr and yr.If type is not "data", then y is approximated using a linear combination of basis functions
f0, f1, f2, ...:y = c0 f0(x) + c1 f1(x) + c2 f2(x) + ...
In this case, the xr and yr input data ranges must be the same size.The result coefficients are stored in cr, so the size of that range determines the number of data points or basis functions to be used in the approximation for each y value.
The return value is the rank of the matrix which is constructed to solve for the coefficients, as determined by a singular value decomposition.
Available basis function types are:
- "poly"
- The polynomial basis functions are:
1, x, x2, x3, ... - (When other basis function types are implemented they will be documented here)
search {xr}=search(f,x0[,dx[,x1...xN]]), search for minimum of f(xr)- Determines values for the N
elements of range xr which minimize the function
defined by the cell or symbol formula f.
Uses the Nelder-Mead simplex algorithm for unconstrained non-linear
function minimization.
x0 specifies the initial value for xr. Optionally, dx specifies step sizes used to create the initial simplex, and x1...xN specifies the initial simplex vertices. If x1...xN are specified then dx is only used for restarts. If dx is not specified then a default step size of +10% in each direction is used. x0, dx, and x1...xN may be expression lists or ranges.
The following variables can be used to modify the tolerances and limits used by search:
symbol name default value description ----------- ------------- ----------- search_func_limit 200*N limit on number of function evaluations search_func_tol 10-4 function tolerance search_size_tol 10-4 simplex size tolerance search_cond_check 5*N how often to check condition number search_cond_limit 104 limit on condition number for restart
The search terminates successfully when the difference between the highest and lowest function values is less than or equal to search_func_tol and the simplex size (a measure of the deviation in the xr values) is less than or equal to search_size_tol. The search terminates unsuccessfully and displays a warning if the number of function evaluations reaches search_func_limit.In some cases the search algorithm can get stuck and converge to a non-minimum point. This behavior can be detected by monitoring the condition number of the matrix of simplex directions, checking it every search_cond_check iterations. If the condition number exceeds search_cond_limit then the search is restarted using orthogonal steps around the current lowest point.
- Search Tips:
- To maximize a function, define the formula to be the
negative of the desired function.
To find a zero of a function, define the formula to be the absolute value of the desired function.
If the formula is defined in a symbol and search produces warnings about cyclic dependencies, try defining the formula in a cell instead.
For debugging, set debug on to cause search to produce detailed output as it proceeds.