The Dynamical Systems package

poly$-$scalar$-$composition

$\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Returns a polynomial obtained by substituting a list of polynomials G=(G1,G2,...,GN) into a polynomial F(X1,X2,...,XN). All polynomials are assumed to be in the internal form, so variables do not explicitly apprear in the calculation.

poly$-$composition

$\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Return the composition of a polynomial map F with a polynomial map G. Both maps are represented as lists of polynomials, and each polynomial is in the internal alist representation. The restriction is that the length of the list G must be the number of variables in the list F.

poly$-$dynamic$-$power

$\parbox{\pboxargslen}{\em f n {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Calculate the composition FoFo...oF (n times), where F is a polynomial map represented as a list of polynomials.

poly$-$scalar$-$evaluate

$\parbox{\pboxargslen}{\em f x {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Evaluate a polynomial F at a point X. This operation is implemented through POLY$-$SCALAR$-$COMPOSITION.

poly$-$evaluate

$\parbox{\pboxargslen}{\em f x {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Evaluate a polynomial map F, represented as list of polynomials, at a point X.

factorial

$\parbox{\pboxargslen}{\em n {\sf \&optional} (k n) {\sf \&aux} (result 1) \/}$ [FUNCTION]

Return N!/(N$-$K)!=N(N$-$1)(N$-$K+1).

poly$-$scalar$-$diff

$\parbox{\pboxargslen}{\em f m \/}$ [FUNCTION]

Return the partial derivative of a polynomial F over multiple variables according to multiindex M.

poly$-$diff

$\parbox{\pboxargslen}{\em f m \/}$ [FUNCTION]

Return the partial derivative of a polynomial map F, represented as a list of polynomials, with respect to several variables, according to multi$-$index M.

standard$-$vector

$\parbox{\pboxargslen}{\em n k {\sf \&optional} (coeff 1) {\sf \&aux} (v (make$-$list n :initial$-$element 0)) \/}$ [FUNCTION]

Returns vector (0 0 ... 1 ... 0 0) of length N, where 1 appears on K$-$th place.

scalar$-$partial

$\parbox{\pboxargslen}{\em f k {\sf \&optional} (l 1) \/}$ [FUNCTION]

Returns the L$-$th partial derivative of a polynomial F over the K$-$th variable.

partial

$\parbox{\pboxargslen}{\em f k {\sf \&optional} (l 1) \/}$ [FUNCTION]

Returns the L$-$th partial derivative over the K$-$th variable, of a polynomial map F, represented as a list of polynomials.

determinant

$\parbox{\pboxargslen}{\em f {\sf \&optional} (order \char93 'lex$\gt$) {\sf \&aux} (result nil) \/}$ [FUNCTION]

Returns the determinant of a polynomial matrix F, which is a list of rows of the matrix, each row is a list of polynomials. The algorithm is recursive expansion along columns.

minor

$\parbox{\pboxargslen}{\em f i j {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Calculate the minor of a polynomial matrix F with respect to entry (I,J).

drop$-$row

$\parbox{\pboxargslen}{\em f i \/}$ [FUNCTION]

Discards the I$-$th row from a polynomial matrix F.

drop$-$column

$\parbox{\pboxargslen}{\em f j \/}$ [FUNCTION]

Discards the J$-$th column from a polynomial matrix F.

drop$-$elt

$\parbox{\pboxargslen}{\em lst j \/}$ [FUNCTION]

Discards the J$-$th element from a list LST.

matrix$-$

$\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Returns difference of two polynomial matrices F and G.

scalar$-$times$-$matrix

$\parbox{\pboxargslen}{\em s f \/}$ [FUNCTION]

Returns a product of a polynomial S by a polynomial matrix F.

monom$-$times$-$matrix

$\parbox{\pboxargslen}{\em m f \/}$ [FUNCTION]

Returns a product of a monomial M by a polynomial matrix F.

term$-$times$-$matrix

$\parbox{\pboxargslen}{\em term f \/}$ [FUNCTION]

Returns a product of a term TERM by a polynomial matrix F.

poly$-$list$-$

$\parbox{\pboxargslen}{\em f g {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Returns the list of differences of two lists of polynomials F and G (polynomial maps).

scalar$-$times$-$poly$-$list

$\parbox{\pboxargslen}{\em s f \/}$ [FUNCTION]

Returns the list of products of a polynomial S by the list of polynomials F.

monom$-$times$-$poly$-$list

$\parbox{\pboxargslen}{\em m f \/}$ [FUNCTION]

Returns the list of products of a monomial M by the list of polynomials F.

term$-$times$-$poly$-$list

$\parbox{\pboxargslen}{\em term f \/}$ [FUNCTION]

Returns the list of products of a term TERM by the list of polynomials F.

characteristic$-$combination

$\parbox{\pboxargslen}{\em a b {\sf \&optional} (order \char93 'lex$\gt$) {\sf \&aux} (n (length b)) \/}$ [FUNCTION]

Returns A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM where A is a polynomial and B=(B1,B2,...,BM) is a polynomial list, where U1, U2, ... , UM are new variables. These variables will be added to every polynomial A and BI as the last M variables.

characteristic$-$combination$-$poly$-$list

$\parbox{\pboxargslen}{\em a b {\sf \&optional} (order \char93 'lex$\gt$) \/}$ [FUNCTION]

Returns A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM where A is a polynomial list and B=(B1, B2, ... , BM) is a list of polynomial lists, where U1, U2, ... ,UM are new variables. These variables will be added to every polynomial A and BI as the last M variables. Se also CHARACTERISTIC$-$COMBINATION.

characteristic$-$matrix

$\parbox{\pboxargslen}{\em a {\sf \&optional} (order \char93 'lex$\gt$) (b (list (identity$-$matrix (length a) (length (caaaar a))))) \/}$ [FUNCTION]

Returns A $-$ U1*B1 $-$ U2*B2 $-$ ... $-$ UM * BM where A is a polynomial matrix and B=(B1,B2,...,BM) is a list of polynomial matrices, where U1, U2, .., UM are new variables. These variables will be added to every polynomial A and BI as the last M variables. Se also CHARACTERISTIC$-$COMBINATION.

characteristic$-$polynomial

$\parbox{\pboxargslen}{\em a {\sf \&optional} (order \char93 'lex$\gt$) (b (list (identity$-$matrix (length a) (length (caaaar a))))) \/}$ [FUNCTION]

Returns the generalized characteristic polynomial, i.e. the determinant DET(A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM), where A and BI are square polynomial matrices in N variables. The resulting polynomial will have N+M variables, with U1, U2, ..., UM added as the last M variables.

identity$-$matrix

$\parbox{\pboxargslen}{\em dim nvars \/}$ [FUNCTION]

Return the polynomial matrix which is the identity matrix. DIM is the requested dimension and NVARS is the number of variables of each entry.

print$-$matrix

$\parbox{\pboxargslen}{\em f vars \/}$ [FUNCTION]

Prints a polynomial matrix F, using a list of symbols VARS as variable names.

jacobi$-$matrix

$\parbox{\pboxargslen}{\em f {\sf \&optional} (m (length f)) (n (length (caaaar f))) \/}$ [FUNCTION]

Returns the Jacobi matrix of a polynomial list F over the first N variables.

jacobian

$\parbox{\pboxargslen}{\em f {\sf \&optional} (order \char93 'lex$\gt$) (m (length f)) (n (length (caaaar f))) \/}$ [FUNCTION]

Returns the Jacobian (determinant) of a polynomial list F over the first N variables.