polynomialIndex: Multivariate Polynomial Representation

Description

Function polynomialIndex provides matrix which allows to iterate through the elements of multivariate polynomial being aware of these elements powers. So (i, j)-th element of the matrix is power of j-th variable in i-th multivariate polynomial element.

Function printPolynomial prints multivariate polynomial given its degrees (pol_degrees) and coefficients (pol_coefficients) vectors.

Usage

polynomialIndex(pol_degrees = numeric(0), is_validation = TRUE)
printPolynomial(pol_degrees, pol_coefficients, is_validation = TRUE)

Value

Function polynomialIndex

returns matrix which rows are responsible for variables while columns are related to powers. So $(i, j)$ -th element of this matrix corresponds to the power $i_{j}$ of the $x_{j}$ variable in $i$ -th polynomial element. Therefore $i$ -th column of this matrix contains vector of powers $(i_{1}, . . ., i_{m})$ for the $i$ -th polynomial element. So the function transforms $m$ -dimensional elements indexing to one-dimensional.

Function printPolynomial returns the string which contains polynomial symbolic representation.

Arguments

pol_degrees: non-negative integer vector of polynomial degrees (orders).
is_validation: logical value indicating whether function input arguments should be validated. Set it to FALSE for slight performance boost (default value is TRUE).
pol_coefficients: numeric vector of polynomial coefficients.

Details

Multivariate polynomial of degrees $(K_{1}, . . ., K_{m})$ (pol_degrees) has the form: $a_{(0, . . ., 0)} x_{1}^{0} * . . . * x_{m}^{0} + . . . + a_{(K_{1}, . . ., K_{m})} x_{1}^{K_{1}} * . . . * x_{m}^{K_{m}},$ where $a_{(i_{1}, . . ., i_{m})}$ are polynomial coefficients, while polynomial elements are: $a_{(i_{1}, . . ., i_{m})} x_{1}^{i_{1}} * . . . * x_{m}^{i_{m}},$ where $(i_{1}, . . ., i_{m})$ are polynomial element's powers corresponding to variables $(x_{1}, . . ., x_{m})$ respectively. Note that $i_{j} \in {0, . . ., K_{j}}$ .

Function printPolynomial removes polynomial elements which coefficients are zero and variables which powers are zero. Output may contain long coefficients representation as they are not rounded.

Examples

Run this code

## Get polynomial indexes matrix for the polynomial 
## which degrees are (1, 3, 5)

polynomialIndex(c(1, 3, 5))
# \donttest{
## Consider multivariate polynomial of degrees (2, 1) such that coefficients
## for elements which powers sum is even are 2 and for those which powers sum
## is odd are 5. So the polynomial is 2+5x2+5x1+2x1x2+2x1^2+5x1^2x2 where
## x1 and x2 are polynomial variables.

# Create variable to store polynomial degrees
pol_degrees <- c(2, 1)

# Let's represent its powers (not coefficients) in a matrix form
pol_matrix <- polynomialIndex(pol_degrees)

# Calculate polynomial elements' powers sums
pol_powers_sum <- pol_matrix[1, ] + pol_matrix[2, ]

# Let's create polynomial coefficients vector filling it
# with NA values
pol_coefficients <- rep(NA, (pol_degrees[1] + 1) * (pol_degrees[2] + 1))

# Now let's fill coefficients vector with correct values
pol_coefficients[pol_powers_sum %% 2 == 0] <- 2
pol_coefficients[pol_powers_sum %% 2 != 0] <- 5

# Finally, let's check that correspondence is correct
printPolynomial(pol_degrees, pol_coefficients)
# }
## Let's represent polynomial 0.3+0.5x2-x2^2+2x1+1.5x1x2+x1x2^2

pol_degrees <- c(1, 2)
pol_coefficients <- c(0.3, 0.5, -1, 2, 1.5, 1)

printPolynomial(pol_degrees, pol_coefficients)