‣ Relu( a ) | ( operation ) |
Returns: a float
The ReLU (Rectified Linear Unit) activation function. Returns Maximum(a, 0) for a float a.
‣ KroneckerDelta( a, b ) | ( function ) |
Returns: 0 or 1
The Kronecker delta function. Returns 1 if a equals b, otherwise 0.
‣ MultiplyMatrices( m_1, mat_1, n_1, m_2, mat_2, n_2 ) | ( function ) |
Returns: a matrix
Multiply two matrices with explicit dimensions. mat_1 is an m_1 x n_1 matrix, mat_2 is an m_2 x n_2 matrix. Requires n_1 = m_2.
‣ Enumerate( l ) | ( operation ) |
Returns: a list of pairs
Create enumerated pairs from a list. Returns a list of pairs [i, l[i]] for each element in l.
‣ SplitDenseList( l, dims ) | ( operation ) |
Returns: a list of lists
Split a dense list l according to the dimensions in dims. The sum of dimensions must equal the length of the list.
‣ SplitDenseList( l, n ) | ( operation ) |
Returns: a list of lists
Split a dense list l into sublists of size n. If the length of l is not a multiple of n, the last sublist will contain the remaining elements (i.e., will contain less than n elements).
‣ SelectBasedOnCondition( cond, val_true, val_false ) | ( function ) |
Returns: an element
Select an element based on a condition. If cond is true, returns val_true, otherwise returns val_false.
‣ CallFuncListBasedOnCondition( cond, func_1, func_2, args ) | ( function ) |
Returns: the result of the called function
Call one of two functions based on a condition. If cond is true, calls func_1, otherwise calls func_2 on the arguments in args. For example, CallFuncListBasedOnCondition( true, x -> x^2, x -> x^3, [ 2 ] ); returns \(4\), while CallFuncListBasedOnCondition( false, x -> x^2, x -> x^3, [ 2 ] ); returns \(8\).
‣ Diff( vars, str, i ) | ( operation ) |
Returns: a function
Compute the partial derivative of an expression with respect to the i-th variable. vars is a list of variable names, str is the expression string. See the example in the Expressions chapter.
‣ LazyDiff( vars, str, i ) | ( operation ) |
Returns: a function
Compute the lazy partial derivative of an expression with respect to the i-th variable. vars is a list of variable names, str is the expression string. See the example in the Expressions chapter.
‣ JacobianMatrixUsingPython( vars, exps, indices ) | ( operation ) |
Returns: a matrix of derivative expressions
Compute the Jacobian matrix using Python's SymPy library. vars is a list of variable names, exps is a list of expression strings, indices specifies which variables to differentiate with respect to.
‣ LazyJacobianMatrix( vars, exps, indices ) | ( operation ) |
Returns: a function
Compute a lazy Jacobian matrix (deferred computation). Returns a function that computes the Jacobian when evaluated.
‣ ScatterPlotUsingPython( points, labels ) | ( operation ) |
Returns: the directory containing the plot
Create a scatter plot using Python's matplotlib. points is a list of 2D points, labels is a list of class labels.
‣ SimplifyExpressionUsingPython( vars, exps ) | ( operation ) |
Returns: a list of simplified expression strings
Simplify expressions using Python's SymPy library. vars is a list of variable names, exps is a list of expression strings.
‣ LaTeXOutputUsingPython( vars, exps ) | ( operation ) |
Returns: a list of LaTeX strings
Convert expressions to LaTeX format using Python's SymPy library. vars is a list of variable names, exps is a list of expression strings.
‣ AsCythonFunction( vars, function_names, functions ) | ( operation ) |
Returns: a string with instructions to use the compiled functions
Compile functions to Cython for improved performance. vars is a list of lists of variable names (one per function), function_names is a list of function names, functions is a list of function body strings.
gap> dummy_input := CreateContextualVariables( [ "a", "b", "c" ] ); [ a, b, c ] gap> AssignExpressions( dummy_input );; #I MakeReadWriteGlobal: a already read-write #I MakeReadWriteGlobal: b already read-write #I MakeReadWriteGlobal: c already read-write gap> e := Sin( a ) + Cos( b ); Sin( a ) + Cos( b ) gap> Diff( e, 1 )( dummy_input ); Cos( a ) gap> LazyDiff( e, 1 )( dummy_input ); Diff( [ "a", "b", "c" ], "(Sin(a))+(Cos(b))", 1 )( [ a, b, c ] ) gap> JacobianMatrixUsingPython( [ a*Cos(b)+Exp(c), a*b*c ], [ 1, 2, 3 ] ); [ [ "Cos(b)", "-a*Sin(b)", "Exp(c)" ], [ "b*c", "a*c", "a*b" ] ] gap> JacobianMatrix( > [ "a", "b", "c" ], > [ "a*Cos(b)+Exp(c)", "a*b*c" ], > [ 1, 2, 3 ] )(dummy_input); [ [ Cos(b), (-a)*Sin(b), Exp(c) ], [ b*c, a*c, a*b ] ] gap> SimplifyExpressionUsingPython( > [ "a", "b" ], > [ "Sin(a)^2 + Cos(a)^2", "Exp(Log(b))" ] ); [ "1", "b" ] gap> LaTeXOutputUsingPython( e ); "\\sin{\\left(a \\right)} + \\cos{\\left(b \\right)}" gap> AsCythonFunction( [[ "x", "y" ], [ "z" ]], ["f", "g"], ["x*y", "Sin(z)"] );; gap> """ > It will produce output similar to the following lines: > $ cd /tmp/gaptempdirI6rq3l/ > $ python > >>> from cython_functions import f, g > >>> w = [ 2, 3 ] # or any other vector in R^2 > >>> f(w) > 6.0 > """;;
gap> sigmoid := Expression( [ "x" ], "Exp(x)/(1+Exp(x))" ); Exp( x ) / (1 + Exp( x )) gap> sigmoid := AsFunction( sigmoid ); function( vec ) ... end gap> sigmoid( [ 0 ] ); 0.5 gap> points := List( 0.1 * [ -20 .. 20 ], x -> [ x, sigmoid( [ x ] ) ] ); [ [ -2., 0.119203 ], [ -1.9, 0.130108 ], [ -1.8, 0.141851 ], [ -1.7, 0.154465 ], [ -1.6, 0.167982 ], [ -1.5, 0.182426 ], [ -1.4, 0.197816 ], [ -1.3, 0.214165 ], [ -1.2, 0.231475 ], [ -1.1, 0.24974 ], [ -1., 0.268941 ], [ -0.9, 0.28905 ], [ -0.8, 0.310026 ], [ -0.7, 0.331812 ], [ -0.6, 0.354344 ], [ -0.5, 0.377541 ], [ -0.4, 0.401312 ], [ -0.3, 0.425557 ], [ -0.2, 0.450166 ], [ -0.1, 0.475021 ], [ 0., 0.5 ], [ 0.1, 0.524979 ], [ 0.2, 0.549834 ], [ 0.3, 0.574443 ], [ 0.4, 0.598688 ], [ 0.5, 0.622459 ], [ 0.6, 0.645656 ], [ 0.7, 0.668188 ], [ 0.8, 0.689974 ], [ 0.9, 0.71095 ], [ 1., 0.731059 ], [ 1.1, 0.75026 ], [ 1.2, 0.768525 ], [ 1.3, 0.785835 ], [ 1.4, 0.802184 ], [ 1.5, 0.817574 ], [ 1.6, 0.832018 ], [ 1.7, 0.845535 ], [ 1.8, 0.858149 ], [ 1.9, 0.869892 ], [ 2., 0.880797 ] ] gap> labels := List( points, point -> 0 );; gap> ScatterPlotUsingPython( points, labels : size := "100", action := "save" );;
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