Version:	0.9.16
Title:	Infrastructure for Manipulation Polynomial Matrices
Description:	Implementation of class "polyMatrix" for storing a matrix of polynomials and implements basic matrix operations; including a determinant and characteristic polynomial. It is based on the package 'polynom' and uses a lot of its methods to implement matrix operations. This package includes 3 methods of triangularization of polynomial matrices: Extended Euclidean algorithm which is most classical but numerically unstable; Sylvester algorithm based on LQ decomposition; Interpolation algorithm is based on LQ decomposition and Newton interpolation. Both methods are described in D. Henrion & M. Sebek, Reliable numerical methods for polynomial matrix triangularization, IEEE Transactions on Automatic Control (Volume 44, Issue 3, Mar 1999, Pages 497-508) <doi:10.1109/9.751344> and in Salah Labhalla, Henri Lombardi & Roger Marlin, Algorithmes de calcule de la reduction de Hermite d'une matrice a coefficients polynomeaux, Theoretical Computer Science (Volume 161, Issue 1-2, July 1996, Pages 69-92) <doi:10.1016/0304-3975(95)00090-9>.
Type:	Package
Imports:	methods, polynom, Matrix
License:	MIT + file LICENSE
Depends:	R (≥ 4.0)
Suggests:	testthat, withr
Repository:	CRAN
URL:	https://github.com/namezys/polymatrix
BugReports:	https://github.com/namezys/polymatrix/issues
RoxygenNote:	7.1.1
NeedsCompilation:	no
Packaged:	2021-07-18 13:48:59 UTC; namezys
Author:	Tamas Prohle [aut], Peter Prohle [aut], Nikolai Ryzhkov [aut, cre], Ildiko Laszlo [aut], Ulas Onat Alakent [ctb]
Maintainer:	Nikolai Ryzhkov <namezys@gmail.com>
Date/Publication:	2021-07-18 14:00:02 UTC

Implementation of matrices of polynomials

Description

Implementation of class "polyMatrix" for storing a matrix of polynomials and implements basic matrix operations; including a determinant and characteristic polynomial. It is based on the package 'polynom' and uses a lot of its methods to implement matrix operations. This package includes 3 methods of triangularization of polynomial matrices: Extended Euclidean algorithm which is most classical but numerically unstable; Sylvester algorithm based on LQ decomposition; Interpolation algorithm is based on LQ decomposition and Newton interpolation. Both methods are described in D. Henrion & M. Sebek, Reliable numerical methods for polynomial matrix triangularization, IEEE Transactions on Automatic Control (Volume 44, Issue 3, Mar 1999, Pages 497-508) <doi:10.1109/9.751344> and in Salah Labhalla, Henri Lombardi & Roger Marlin, Algorithmes de calcule de la reduction de Hermite d'une matrice a coefficients polynomeaux, Theoretical Computer Science (Volume 161, Issue 1-2, July 1996, Pages 69-92) <doi:10.1016/0304-3975(95)00090-9>.

Details

This package add transperent support of matrices of polynomials to R language.

Input

• low level input of matrix: polyMatrix()

• string ibpuy of polynomial: parse.polynomial()

• string input of matrix: parse.polyMatrix()

Output

Matrices of polynomials uses same format as numerical matrices

Operators

Basic arithmetic operations were implemented.

Functions

The most important matrix functions are implemented:

• the determinant det()

Author(s)

Maintainer: Nikolai Ryzhkov namezys@gmail.com (ORCID)

Authors:

• Tamas Prohle prohlet@ludens.elte.hu

• Peter Prohle prohlep@math.elte.hu

• Ildiko Laszlo (ORCID)

Other contributors:

• Ulas Onat Alakent [contributor]

See Also

Useful links:

• https://github.com/namezys/polymatrix

• Report bugs at https://github.com/namezys/polymatrix/issues

Matrix multiplication

Description

Matrix multiplication accepts both polynomial and numerical matrices.

Usage

## S4 method for signature 'polyMatrix,polyMatrix'
x %*% y

## S4 method for signature 'polyMatrix,matrix'
x %*% y

## S4 method for signature 'matrix,polyMatrix'
x %*% y

Arguments

GCD for polynomial matrices

Description

The greatest common divisor of polynomials or polynomial matrices.

Usage

GCD(...)

## S4 method for signature 'polyMatrix'
GCD(...)

Arguments

Methods (by class)

• polyMatrix: the greatest common divisor of all elements of the polynomial matrix

See Also

polynomial implementation polynom::GCD() and LCM()

Examples

# GCD of polynomial matrix

GCD(parse.polyMatrix(
 "  1 - x, 1 - x^2, 1 + 2*x + x^2",
 "x - x^2,   1 + x, 1 - 2*x + x^2"
))  ## 1

LCM for polynomial matrices

Description

The least common multiple of polynomials or polynomial matrices.

Usage

LCM(...)

## S4 method for signature 'polyMatrix'
LCM(...)

Arguments

Methods (by class)

• polyMatrix: the least common multiple of polynomial matrices

See Also

polynomial implementation polynom::GCD() and GCD()

Examples

# LCM of polynomial matrix
LCM(parse.polyMatrix(
 "  1 - x, 1 - x^2, 1 + 2*x + x^2",
 "x - x^2,   1 + x, 1 - 2*x + x^2"
))  ## 0.25*x - 0.5*x^3 + 0.25*x^5

Extract or Replace Parts of a polynomial matrix

Description

Extract or Replace Parts of a polynomial matrix

Usage

## S4 method for signature 'polyMatrix,missing,missing,missing'
x[i, j, ..., drop = TRUE]

## S4 method for signature 'polyMatrix,missing,ANY,missing'
x[i, j]

## S4 method for signature 'polyMatrix,ANY,missing,missing'
x[i, j]

## S4 method for signature 'polyMatrix,logical,logical,missing'
x[i, j]

## S4 method for signature 'polyMatrix,logical,numeric,missing'
x[i, j]

## S4 method for signature 'polyMatrix,numeric,logical,missing'
x[i, j]

## S4 method for signature 'polyMatrix,numeric,numeric,missing'
x[i, j]

## S4 replacement method for signature 'polyMatrix,missing,missing,ANY'
x[i, j] <- value

## S4 replacement method for signature 'polyMatrix,missing,ANY,ANY'
x[i, j] <- value

## S4 replacement method for signature 'polyMatrix,ANY,missing,ANY'
x[i, j] <- value

## S4 replacement method for signature 'polyMatrix,numeric,numeric,numeric'
x[i, j] <- value

## S4 replacement method for signature 'polyMatrix,numeric,numeric,matrix'
x[i, j] <- value

## S4 replacement method for signature 'polyMatrix,numeric,numeric,polynomial'
x[i, j] <- value

## S4 replacement method for signature 'polyMatrix,numeric,numeric,polyMatrix'
x[i, j] <- value

Arguments

Functions

• [,polyMatrix,missing,ANY,missing-method: get columns

• [,polyMatrix,ANY,missing,missing-method: gets rows

• [,polyMatrix,logical,logical,missing-method: gets by logical index

• [,polyMatrix,logical,numeric,missing-method: gets by logical index and numerical indices

• [,polyMatrix,numeric,logical,missing-method: gets by logical index and numerical indices

• [,polyMatrix,numeric,numeric,missing-method: gets by row and column indices

• [<-,polyMatrix,missing,missing,ANY-method: replace o matrix by a new one

• [<-,polyMatrix,missing,ANY,ANY-method: assigns rows

• [<-,polyMatrix,ANY,missing,ANY-method: assigns columns

• [<-,polyMatrix,numeric,numeric,numeric-method: replace part of matrix by one number

• [<-,polyMatrix,numeric,numeric,matrix-method: replace part of matrix by another numerical matrix. Size of the new matrix should be same as replaced part

• [<-,polyMatrix,numeric,numeric,polynomial-method: replace part of matrix by one polynomail

• [<-,polyMatrix,numeric,numeric,polyMatrix-method: replace part of matrix by another polunomial matrix. Size of the new matrix should be same as replaced part

Adjungate or classical adjoint of a square matrix

Description

The adjungate or classical adjoint of a square matrix is the transpose of its cofactor matrix. It is also occasionally known as adjunct matrix, though this nomenclature appears to have been decreased in usage.

Usage

adjoint(x)

## S4 method for signature 'polyMatrix'
adjoint(x)

Arguments

Methods (by class)

• polyMatrix: adjungate of polynomial matrix DON'T UNDERSTAND!!!

Combine polynomial matrices by rows or columns

Description

Combine polynomial matrices by rows or columns

Usage

cbind(..., deparse.level = 1)

rbind(..., deparse.level = 1)

Arguments

Value

if at least one argument is a polynomial matrix, the result will be a combined polynomial matrix. Otherwise, the base package implementation base::cbind() or base::rbind() will be called.

Functions

• rbind: row based bind DON'T UNDERSTAND.. !!!

See Also

base::cbind()

Characteristic polynomial of a matrix

Description

Characteristic polynomial of a matrix

Usage

charpolynom(x)

## S4 method for signature 'matrix'
charpolynom(x)

## S4 method for signature 'polynomial'
charpolynom(x)

## S4 method for signature 'polyMatrix'
charpolynom(x)

## S4 method for signature 'polyMatrixCharPolynomial,ANY'
x[[i]]

## S4 method for signature 'polyMatrixCharPolynomial'
degree(x)

## S4 method for signature 'polyMatrixCharPolynomial'
predict(object, newdata)

## S4 method for signature 'polyMatrixCharPolynomial'
show(object)

Arguments

Details

The characteristic polynom of a polynomial matrix is a polynom with polynomial coefficients.

Value

When the input is a numerical matrix of matrix class the value is a polynomial object.

When the input is a polyMatrix object then the value is polyMatrixCharClass class object,

Methods (by class)

• matrix: for numerical matrix it is a polynomial with numerical coefficients

• polynomial: for polynomial it treats as a matrix 1x1

• polyMatrix: for polynomial matrix has polynomial coefficients

• x = polyMatrixCharPolynomial,i = ANY: get polynomial coefficient of characteristic polynomial

• polyMatrixCharPolynomial: the degree of char polynomial of polynomial matrix

• polyMatrixCharPolynomial: the value of char polynomial in a polynomial point

• polyMatrixCharPolynomial: prints out a text representation of a characteristic polynomial of a polynomial matrix

See Also

polyMatrixCharClass

Examples

# numerical matrices
m <- matrix(c(2, 1,
             -1, 0), 2, 2, byrow=TRUE)
charpolynom(m)

Cofactor of a matrix

Description

Cofactor of a matrix

Usage

cofactor(x, r, c)

Arguments

Value

cofactor which is a number or a polynomial

See Also

adjoint()

Gets the maximum degree of polynomial objects

Description

Returns the maximum degree as an integer number.

Usage

degree(x)

## S4 method for signature 'numeric'
degree(x)

## S4 method for signature 'matrix'
degree(x)

## S4 method for signature 'polynomial'
degree(x)

## S4 method for signature 'polyMatrix'
degree(x)

Arguments

Details

By default, this function raises error for unknown type of object.

A numerical scalar can be treated as a polynomial with zero degree.

A numerical matrix has zero degree as each of its items has zero degree as well.

For polynomials this function returns the highest degree of its terms with non-zero coefficient.

Value

The value is an integer number which can be different from zero only for polynomial objects.

Methods (by class)

• numeric: a scalar argument always has zero degree

• matrix: a numerical matrix always has zero degree

• polynomial: the degree of a polynomial

• polyMatrix: the degree of a polynomial matrix is the highest degree of its elements

Examples

# numerical
degree(1)  ## 0


# numerical matrix
degree(matrix(1:6, 3, 2)) ## 0


# polinomial
degree(parse.polynomial("1")) ## 0
degree(parse.polynomial("1 + x")) ## 1
degree(parse.polynomial("1 + x^3")) ## 3


# polynomial matrices
degree(parse.polyMatrix(
   "x; x^2 + 1",
   "0; 2x"))
## 2

Polynomial matrix Diagonals Extract or construct a diagonal polynomial matrix.

Description

Polynomial matrix Diagonals Extract or construct a diagonal polynomial matrix.

Usage

diag(x = 1, nrow, ncol, names = TRUE)

## S4 method for signature 'polynomial'
diag(x, nrow, ncol)

## S4 method for signature 'polyMatrix'
diag(x)

Arguments

Details

In case of polynomial objects, diag has 2 distinct usage:

• x is a polynomial, returns a polynomial matrix the given diagonal and zero off-diagonal entries.

• x is a polynomial matrix, returns a vector as a polynomial matrix of diagonal elements

For polynomial, either nrow or ncol must be provided.

Methods (by class)

• polynomial: for a polynomial, returns a polynomial matrix with the given diagonal

• polyMatrix: for a polynomial matrix extracts diagonal

  For polynomial matrix, neither nrow nor ncol cannot be provided.

See Also

Base base::diag() for numericals and numerical matrices

Examples

# numericals and numerical matrices
diag(matrix(1:12, 3, 4)) ## 1 5 8
diag(9, 2, 2)
##      [,1] [,2]
## [1,]    9    0
## [2,]    0    9



# polynomial
diag(parse.polynomial("1+x+3x^2"), 2, 3)
##                [,1]           [,2]   [,3]
## [1,]   1 + x + 3x^2              0      0
## [2,]              0   1 + x + 3x^2      0


# polynomial matrix
diag(parse.polyMatrix(
  "-3 + x^2, 2 + 4 x,  -x^2",
  "       1,       2, 3 + x",
  "      2x,       0, 2 - 3x"
))
##            [,1]   [,2]     [,3]
## [1,]   -3 + x^2      2   2 - 3x

Inverse polynomial matrix

Description

During inversion we will try to round elememnts to zero.

Usage

inv(x, eps = ZERO_EPS)

Arguments

Details

Right now only matrices with numerical determinant are supported.

Check if object is polyMatrix

Description

Check if object is polyMatrix

Usage

is.polyMatrix(x)

Arguments

Value

TRUE if object is a polynomial matrix

Examples

is.polyMatrix(c(1, 2, 3))
is.polyMatrix(polyMatrix(0, 2, 2))

Proper polynomial matrices

Description

Tests the proper property of a polynomial matrix. A polynomial matrix is proper if the associeted matrix has a full rank.

Usage

is.proper(pm)

is.column.proper(pm)

is.row.proper(pm)

Arguments

Details

A polynomial matrix is column (row, full) proper (or reduced) if the associated matrix has the same rank as the number of columns (rows)

Value

True if object pm is a (row-/column-) proper matrix

Functions

• is.column.proper: tests if its argument is a column-proper matrix

• is.row.proper: tests if its argument is a row-proper matrix

Examples

pm <- parse.polyMatrix(
  "-1 + 7x     , x",
  " 3 - x + x^2, -1 + x^2 - 3 x^3"
)
is.column.proper(pm)
is.row.proper(pm)
is.proper(pm)

Tests if something is zero or not

Description

Generic function to check if we can treat on object as being zero. For matrices the result is a matrix of the same size.

Usage

is.zero(x, eps = ZERO_EPS)

## S4 method for signature 'polynomial'
is.zero(x, eps = ZERO_EPS)

## S4 method for signature 'polyMatrix'
is.zero(x, eps = ZERO_EPS)

Arguments

Details

Different type of objects can be treated as zero in different ways:

• Numerical types can be compared by absolute value with eps.

• Other types should define its own method.

By befault eps = ⁠{r} ZERO_EPS⁠

Value

TRUE if the object can be treat as zero

Methods (by class)

• polynomial: a polynomial can be treated as zero if all its coefficients can be treated as zero

• polyMatrix: for a polynomial matrix every item is checked if it is zero polynomial

See Also

zero.round()

Examples

# numericals and matrices
is.zero(0)  ## TRUE

is.zero(0.0001, eps=0.01)  ## TRUE

is.zero(c(0, 1, 0))  ## TRUE, FALSE, TRUE

is.zero(matrix(c(1, 9, 0, 0), 2, 2))
## FALSE TRUE
## FALSE TRUE


# polynomials
is.zero(parse.polynomial("0.1 - 0.5 x")) ## FALSE
is.zero(parse.polynomial("0.0001 - 0.0005 x + 0.00002 x^2"), eps=0.01) ## TRUE

Degree of each item of the matrix

Description

Returns a matrix obtained by applying a function degree() for each element of the matrix.

Usage

matrix.degree(x)

## S4 method for signature 'matrix'
matrix.degree(x)

## S4 method for signature 'polynomial'
matrix.degree(x)

## S4 method for signature 'polyMatrix'
matrix.degree(x)

Arguments

Details

Degree of each item is calculated using degree() which is defined for polynomials as the highest degree of the terms with non-zero coefficients.

For convenience this function is defined for any object, but returns zero for non polynomial objects.

Value

If the argument is a matrix, the result is a matrix of the same size containing the degrees of the matrix items.

For a numerical matrix the value is always a zero matrix of the same size

For a polynomial the value is the degree of the polynomial

Methods (by class)

• matrix: the degree of a numerical matrix is a zero matrix for compatibility

• polynomial: the degree of a polynomial

• polyMatrix: a matrix of degrees for each polynomial item of the source matrix

Examples

# numerical matrices
matrix.degree(matrix(1:6, 2, 3))
##      [,1] [,2] [,3]
## [1,]    0    0    0
## [2,]    0    0    0

# polynomials
matrix.degree(parse.polynomial("x + 1")) ## 1
matrix.degree(parse.polynomial("x^3 + 1")) ## 3
matrix.degree(parse.polynomial("1")) ## 0

# polynomial matrices
matrix.degree(parse.polyMatrix(
   "x; x^2 + 1",
   "0; 2x"))
##      [,1] [,2]
## [1,]    1    2
## [2,]    0    1

Minor of matrix item

Description

A minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors).

Usage

minor(x, r, c)

Arguments

Build matrix of polynimal decomposition using Newton interpolation in Newton bais: (x-x_0), (x - x_0) * (x x_1)

Description

Build matrix of polynimal decomposition using Newton interpolation in Newton bais: (x-x_0), (x - x_0) * (x x_1)

Usage

newton(C, points)

Arguments

Value

Matrix of coefficients in columns (from higher degree to lower)

Parse polynomial matrix from strings

Description

This is a convenient way to input a polynomial matrix.

Usage

parse.polyMatrix(..., var = "x")

Arguments

Details

Space and tabulation characters are ignored.

Row should be divided by new line "\n" or backslash "\" (TeX style).

Elements in each row can be divided by ",", ";" or "&" (TeX style)

For convenience, this function can accept multiple string. In this case each string will be treated as a new row.

This function accepts TeX matrix format.

Value

new polynomial matrix of polyMatrix class

See Also

parse.polynomial()

Examples

parse.polyMatrix("       1, 2 + x",
                 "2 + 2x^2,    x^3")

# The function can suggest mistake position in case of invalid format
## Not run: 
parse.polyMatrix(
    "1 + y &    2\\
        -2 &  x^2"
)
## Fail to parse polyMatrix: invalid term at position 2 in item [1, 1]

## End(Not run)

Parse polynomial from string

Description

Parse string representation of polynomial into a polynomial object.

Usage

parse.polynomial(s, var = "x")

Arguments

Value

new polynomial as polynom::polynomial object

See Also

parse.polyMatrix()

Create polyMatrix object

Description

This function will create a polynomial object from coefficient matrix or signle value

Usage

polyMatrix(data, nrow, ncol, degree)

Arguments

Details

A coefficient matrix is a matrix which contains matrices of coefficients starting from lower degree to higher ones, side-by-side

Value

new polynomial matrix of polyMatrix class

Arithmetic Operators

Description

These unary and binary operators perform arithmetical operations on polynomial or numerical marices.

Usage

## S4 method for signature 'polyMatrix,missing'
e1 + e2

## S4 method for signature 'polyMatrix,polyMatrix'
e1 + e2

## S4 method for signature 'polyMatrix,polynomial'
e1 + e2

## S4 method for signature 'polyMatrix,numeric'
e1 + e2

## S4 method for signature 'polyMatrix,matrix'
e1 + e2

## S4 method for signature 'ANY,polyMatrix'
e1 + e2

## S4 method for signature 'polyMatrix,numeric'
e1 * e2

## S4 method for signature 'polyMatrix,polynomial'
e1 * e2

## S4 method for signature 'polyMatrix,polyMatrix'
e1 * e2

## S4 method for signature 'ANY,polyMatrix'
e1 * e2

## S4 method for signature 'polyMatrix,polyMatrix'
e1 - e2

## S4 method for signature 'polyMatrix,ANY'
e1 - e2

## S4 method for signature 'ANY,polyMatrix'
e1 - e2

Arguments

Details

Both operands can be:

• numerical scalar

• polynomial scalar

• numerical matrix

• polynomial matrix

Value

Unary + return same object.

Binary + with two matrix operands returns elementwise summation.

Binary + with matrix and scalar operands returns elementwise summation with scalar.

Binary * is elementwise multiplication with matrix or scalar operands.

Unary - return a matrix with changed sign.

Binary '-' of matrices or scalar operands returns matrix subtraction.

Functions

• +,polyMatrix,missing-method: unary +

• -,polyMatrix,polyMatrix-method: unary -

A class to represent a matrix of polynomials

Description

A class to represent a matrix of polynomials

Usage

## S4 method for signature 'polyMatrix,numeric'
x[[i]]

## S4 method for signature 'polyMatrix'
det(x)

## S4 method for signature 'polyMatrix'
nrow(x)

## S4 method for signature 'polynomial'
nrow(x)

## S4 method for signature 'polyMatrix'
ncol(x)

## S4 method for signature 'polynomial'
ncol(x)

## S4 method for signature 'polyMatrix'
dim(x)

## S4 method for signature 'polyMatrix'
predict(object, newdata)

## S4 method for signature 'polyMatrix'
round(x, digits = 0)

## S4 method for signature 'polyMatrix'
show(object)

## S4 method for signature 'polyMatrix,polyMatrix'
e1 == e2

## S4 method for signature 'polyMatrix,polynomial'
e1 == e2

## S4 method for signature 'polyMatrix,matrix'
e1 == e2

## S4 method for signature 'polyMatrix,numeric'
e1 == e2

## S4 method for signature 'ANY,polyMatrix'
e1 == e2

## S4 method for signature 'polyMatrix,ANY'
e1 != e2

## S4 method for signature 'ANY,polyMatrix'
e1 != e2

Arguments

Methods (by generic)

• [[: get coefficient matrix by degree

• det: determinant of a polynomial matrix

• nrow: the number of rows of a polynomial matrix

• nrow: a polynomial has only one row

• ncol: the number of columns of a polynomial matrix

• ncol: a polynomial has only one column

• dim: the dimension of a polynomial matrix

• predict: the value of a polynomial matrix in a point

• round: rounding of a polynomial matrix is rounding of polynomial coefficients

• show: prints out a text representation of a polynomial matrix

• ==: equal operator for two polinomial matrices, result is a boolean matrix

• ==: equal operator for polinomail matrix and polinomail, result is a matrix

• ==: equal operator for polinomial and numerical matrices

• ==: equal operator for polinomial matrix and number, result is a matrix

• ==: equal operator for aby object and polinomial matrix

• !=: not equal operator

• !=: not equal operator

Slots

coef

A matrix of coefficients which are joined into one matrix from lower degree to higher

ncol

The actual number of columns in the polynomial matrix

Examples

# create a new polynomial matrix by parsing strings
pm <- parse.polyMatrix(
     "x; 1 + x^2; 3 x - x^2",
     "1; 1 + x^3; - x + x^3"
)

# get coefficient matrix for degree 0
pm[[0]]
##      [,1] [,2] [,3]
## [1,]    0    1    0
## [2 ]    1    1    0
# get coefficient matrix for degree 1
pm[[1]]
##      [,1] [,2] [,3]
## [1,]    1    0    3
## [2 ]    0    0   -1


# dimensions
nrow(pm) ## 2


ncol(pm) ## 3


dim(pm) ## [1] 2 3


# round
round(parse.polyMatrix(
  "      1.0001 - x,            1 - x^2, 1 + 2.0003*x + x^2",
  "0.0001 + x - x^2, 1 + x + 0.0001 x^2, 1 - 2*x + x^2"
))
##           [,1]      [,2]           [,3]
## [1,]     1 - x   1 - x^2   1 + 2x + x^2
## [2,]   x - x^2     1 + x   1 - 2x + x^2


# print out a polynomial matrix
show(parse.polyMatrix(
  "      1.0001 - x,          1 - x^2, 1 + 2.0003*x + x^2",
  "0.0001 + x - x^2,            1 + x, 1 - 2*x + x^2",
  "        12.3 x^3,  2 + 3.5 x + x^4, -0.7 + 1.6e-3 x^3"
))
##                   [,1]             [,2]                [,3]
## [1,]        1.0001 - x          1 - x^2   1 + 2.0003x + x^2
## [2,]   1e-04 + x - x^2            1 + x        1 - 2x + x^2
## [3,]           12.3x^3   2 + 3.5x + x^4    -0.7 + 0.0016x^3

Apply for polynomial matrix

Description

Apply function to each element of matrix

Usage

polyMatrix.apply(x, f)

Arguments

A class to repesent characteristic polynomial of a polynomial matrix

Description

Characteristic polynomial of a polynomial matrix is a polynomial with polynomial coefficients

Polynomial matrix transpose

Description

Given a polyMatrix, t returns the transpose of x

Usage

## S4 method for signature 'polyMatrix'
t(x)

Arguments

See Also

base::t() for numerical matrix tranpose

Examples

pm <- parse.polyMatrix("1, x, x^2",
                       "x, 1, x^3")
t(pm)
##        [,1]   [,2]
## [1,]      1      x
## [2,]      x      1
## [3,]    x^2    x^3

Trace of a 'matrix' or 'polyMatrix' class matrix

Description

Trace of a matrix is the sum of the diagonal elements of the given matrix.

Usage

tr(x)

Arguments

Details

If the given matrix is a polynomial matrix, the result will be a polynomial.

Value

Returns the trace of the given matrix as a number or a polynomial.

Examples

# numerical matrices
m <- matrix(1:12, 3, 4)
##      [,1] [,2] [,3] [,4]
## [1,]    1    4    7   10
## [2,]    2    5    8   11
## [3,]    3    6    9   12
tr(m)  ## 15

# polynomial matrix
pm <- parse.polyMatrix(
  "-3 + x^2, 2 + 4 x,  -x^2",
  "       1,       2, 3 + x",
  "     2*x,       0, 2 - 3 x"
)
tr(pm)  ## 1 - 3*x + x^2

Triangularization of a polynomial matrix by interpolation method

Description

The parameters point_vector, round_digits can significantly affect the result.

Usage

triang_Interpolation(
  pm,
  point_vector,
  round_digits = 5,
  eps = .Machine$double.eps^0.5
)

Arguments

Details

Default value of 'eps“ usually is enought to determintate real zeros.

In a polynomial matrix the head elements are the first non-zero polynomials of columns. The sequence of row indices of this head elements form the shape of the polynomial matrix. A polynomial matrix is in left-lower triangular form, if this sequence is monoton increasing.

This method offers a solution of the triangulrization by the Interpolation method, described in the article of Labhalla-Lombardi-Marlin (1996).

Value

Tranfortmaiton matrix

Triangularization of a polynomial matrix by Sylvester method

Description

The function triang_Sylvester triangularize the given polynomial matrix.

Usage

triang_Sylvester(pm, u, eps = ZERO_EPS)

Arguments

Details

The u parameter is a necessary supplementary input without default value. This parameter give the minimal degree of the searched triangulizator to solve the problem.

In a polynomial matrix the head elements are the first non-zero polynomials of columns. The sequence of row indices of this head elements form the shape of the polynomial matrix. A polynomial matrix is in left-lower triangular form, if this sequence is monoton increasing.

This method search a solution of the triangulrization by the method of Sylvester matrix, descripted in the article Labhalla-Lombardi-Marlin (1996).

Value

T - the left-lower triangularized version of the given polynomial matrix U - the right multiplicator to triangularize the given polynomial matrix

References

Salah Labhalla, Henri Lombardi, Roger Marlin: Algorithm de calcule de la reduction de Hermite d'une matrice a coefficients polynomiaux, Theoretical Computer Science 161 (1996) pp 69-92

Rounds objects to zero if there is too small

Description

Rounds objects to zero if there is too small

Usage

zero.round(x, eps = ZERO_EPS)

## S4 method for signature 'polynomial'
zero.round(x, eps = ZERO_EPS)

## S4 method for signature 'polyMatrix'
zero.round(x, eps = ZERO_EPS)

Arguments

Details

By befault eps = ⁠{r} ZERO_EPS⁠

Methods (by class)

• polynomial: rounding of a polynomial means rounding of each coefficient

• polyMatrix: rounding of a polynomial matrix

See Also

is.zero()

Examples

# numerical
zero.round(1)  ## 1
zero.round(0)  ## 0
zero.round(0.1, eps=0.5) ## 0
zero.round(c(1, 0, .01, 1e-10)) ##  1.00 0.00 0.01 0.00


# polynomials
zero.round(parse.polynomial("0.1 + x + 1e-7 x^2")) ## 0.1 + x
zero.round(parse.polynomial("0.1 + x + 1e-7 x^2"), eps=0.5) ## x


# polynomial matrix
zero.round(parse.polyMatrix(
  "1 + 0.1 x, 10 + x + 3e-8 x^2, 1e-8",
  "0.1 + x^2,     .1 + 1e-8 x^4, 1e-8 x^5"
))
##             [,1]     [,2]   [,3]
## [1,]    1 + 0.1x   10 + x      0
## [2,]   0.1 + x^2      0.1      0

zero.round(parse.polyMatrix(
  "1 + 0.1 x, 10 + x + 3e-8 x^2, 1e-8",
  "0.1 + x^2,     .1 + 1e-8 x^4, 1e-8 x^5"
), eps=0.5)
##        [,1]     [,2]   [,3]
## [1,]      1   10 + x      0
## [2,]    x^2        0      0

Get zero lead hyper rows of size sub_nrow of matrix M

Description

Get zero lead hyper rows of size sub_nrow of matrix M

Usage

zero_lead_hyp_rows(M, sub_nrow, esp = ZERO_EPS)

Arguments

Value

vector of idx of hyperrows, NaN for columns without zeros

Get zero lead rows of matrix M

Description

Get zero lead rows of matrix M

Usage

zero_lead_rows(M, eps = ZERO_EPS)

Arguments

Value

vector of idx (length is equal to columm number), NULL in case of error