Package 'caracas'

Title: Computer Algebra
Description: Computer algebra via the 'SymPy' library (<https://www.sympy.org/>). This makes it possible to solve equations symbolically, find symbolic integrals, symbolic sums and other important quantities.
Authors: Mikkel Meyer Andersen [aut, cre, cph], Søren Højsgaard [aut, cph]
Maintainer: Mikkel Meyer Andersen <[email protected]>
License: GPL
Version: 2.1.1
Built: 2024-10-26 04:56:38 UTC
Source: https://github.com/r-cas/caracas

Help Index

Extract or replace parts of an object

Description

Extract or replace parts of an object

Usage

## S3 method for class 'caracas_symbol'
x[i, j, ..., drop = TRUE]

Arguments

x

A caracas_symbol.
i

row indices specifying elements to extract or replace
j

column indices specifying elements to extract or replace
...

Not used
drop

Simplify dimensions of resulting object

Examples

if (has_sympy()) {
  A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
  B <- as_sym(A)
  B[1:2, ]
  B[, 2]
  B[2, , drop = FALSE]
}

Extract or replace parts of an object

Description

Extract or replace parts of an object

Usage

## S3 replacement method for class 'caracas_symbol'
x[i, j, ...] <- value

Arguments

x

A caracas_symbol.
i

row indices specifying elements to extract or replace
j

column indices specifying elements to extract or replace
...

Not used
value

Replacement value

Examples

if (has_sympy()) {
  A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
  B <- as_sym(A)
  B[, 2] <- "x"
  B[, 3] <- vector_sym(3)
  B
}

Pipe

Description

Pipe operator

Arguments

lhs, rhs

specify what lhs and rhs are

Add prefix to each element of matrix

Description

Add prefix to each element of matrix

Usage

add_prefix(x, prefix = "")

Arguments

x

Numeric or symbolic matrix
prefix

A character vector

Examples

if (has_sympy()) {
  X <- matrix_sym(2, 3)
  X
  add_prefix(X, "e")
  
  X <- matrix(1:6, 3, 2)
  X
  add_prefix(X, "e")
}

All variables

Description

Return all variables in caracas symbol

Usage

all_vars(x)

Arguments

x

caracas symbol

Examples

if (has_sympy()){
  x <- vector_sym(5)
  all_vars(x)
}

Partial fraction decomposition on a rational function

Description

apart() performs a partial fraction decomposition on a rational function

Usage

apart(x)

Arguments

x

A caracas_symbol

Examples

if (has_sympy()){
  def_sym(x)
  expr = (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)
  apart(expr)
}

Coerce symbol to character

Description

Coerce symbol to character

Usage

as_character(x)

Arguments

x

caracas symbol

Get matrix as character matrix

Description

Get matrix as character matrix

Usage

as_character_matrix(x)

Arguments

x

caracas symbol

Examples

if (has_sympy()) {
  s  <- as_sym("[[r1, r2, r3], [u1, u2, u3]]")
  s2 <- apply(as_character_matrix(s), 2, function(x) (paste("1/(", x, ")")))
  as_sym(s2)
}

Construct diagonal matrix from vector

Description

Construct diagonal matrix from vector

Usage

as_diag(x)

Arguments

x

Matrix with 1 row or 1 column that is the diagonal in a new diagonal matrix

Examples

if (has_sympy()) {
  d <- as_sym(c("a", "b", "c"))
  D <- as_diag(d)
  D
}

Convert caracas object to R

Description

Potentially calls doit().

Usage

as_expr(x, first_doit = TRUE)

## S3 method for class 'caracas_symbol'
as.expression(x, ...)

## S3 method for class 'caracas_solve_sys_sol'
as.expression(x, ...)

Arguments

x

caracas_symbol
first_doit

Try doit() first
...

not used

Examples

if (has_sympy()) {
  v <- vector_sym(2)
  x <- as_expr(v)
  x
  y <- as.expression(v)
  y
}

Convert expression into function object.

Description

Convert expression into function object.

Usage

as_func(x, order = NULL, vec_arg = FALSE)

## S3 method for class 'caracas_symbol'
as.function(x, ...)

Arguments

x

caracas expression.
order

desired order of function argument. Defaults to alphabetical ordering.
vec_arg

should the function take vector valued argument.
...

not used

Examples

if (has_sympy()) {
  def_sym(b0, b1, b2, k, x)
  e <- b1 + (b0 - b1)*exp(-k*x) + b2*x

  f1 <- as_func(e)
  f1
  f1(1, 2, 3, 4, 5)
  f1 <- as_func(e, order = sort(all_vars(e)))
  f1(1, 2, 3, 4, 5)
  f2 <- as_func(e, vec_arg = TRUE)
  f2
  f2(c(1, 2, 3, 4, 5))
  f2 <- as_func(e, order = sort(all_vars(e)), vec_arg = TRUE)
  f2
  f2(c(1,2,3,4,5))
  
  f1a <- as.function(e)
  f1a
  f1a(1, 2, 3, 4, 5)
  f1(1, 2, 3, 4, 5)
}

Convert R object to caracas symbol

Description

Variables are detected as a character followed by a number of either: character, number or underscore.

Usage

as_sym(x, declare_symbols = TRUE)

Arguments

x

R object to convert to a symbol
declare_symbols

declare detected symbols automatically

Details

Default is to declare used variables. Alternatively, the user must declare them first, e.g. by symbol().

Note that matrices can be defined by specifying a Python matrix, see below in examples.

Examples

if (has_sympy()) {
  x <- symbol("x")
  A <- matrix(c("x", 0, 0, "2*x"), 2, 2)
  A
  B <- as_sym(A)
  B
  2 * B
  dim(B)
  sqrt(B)
  D <- as_sym("[[1, 4, 5], [-5, 8, 9]]")
  D
}

Stacks matrix to vector

Description

Stacks matrix to vector

Usage

as_vec(x)

Arguments

x

Matrix

Examples

if (has_sympy()) {
  A <- as_sym(matrix(1:9, 3))
  as_vec(A)
}

Convert symbol to character

Description

Convert symbol to character

Usage

## S3 method for class 'caracas_symbol'
as.character(x, replace_I = TRUE, ...)

Arguments

x

A caracas_symbol
replace_I

Replace constant I (can both be identity and imaginary unit)
...

not used

Ask for a symbol's property

Description

Ask for a symbol's property

Usage

ask(x, property)

Arguments

x

symbol
property

property, e.g. 'positive'

Examples

if (has_sympy()) {
  x <- symbol("x", positive = TRUE)
  ask(x, "positive")
}

Put rational function into standard form

Description

cancel() will take any rational function and put it into the standard canonical form

Usage

cancel(x)

Arguments

x

A caracas_symbol

Examples

if (has_sympy()){
  def_sym(x, y, z)
  expr = cancel((x**2 + 2*x + 1)/(x**2 + x))
  cancel(expr)
  expr = (x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z + z**2)/(x**2 - 1)
  cancel(expr)
  factor_(expr)
}

Collects common powers of a term in an expression

Description

Collects common powers of a term in an expression

Usage

collect(x, a)

Arguments

x, a

A caracas_symbol

Examples

if (has_sympy()){
  def_sym(x, y, z)
  expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3
  collect(expr, x)
}

Column space (range) of a symbolic matrix

Description

Column space (range) of a symbolic matrix

Usage

colspan(x)

Arguments

x

Symbolic matrix

Examples

if (has_sympy()) {
  X1 <- matrix_(paste0("x_",c(1,1,1,1, 2,2,2,2, 3,4,3,4)), nrow = 4)
  X1
  colspan(X1)
  do_la(X1, "columnspace")
  rankMatrix_(X1)
  
  X2 <- matrix_(paste0("x_",c(1,1,1,1, 0,0,2,2, 3,4,3,4)), nrow = 4)
  X2
  colspan(X2)
  do_la(X2, "columnspace")
  rankMatrix_(X2)
}

Cumulative Sums

Description

Cumulative Sums

Usage

## S3 method for class 'caracas_symbol'
cumsum(x)

Arguments

x

Elements to sum

Examples

if (has_sympy()) {
  A <- matrix(1:9, 3)
  cumsum(A)
  B <- matrix_sym(3, 3)
  cumsum(B)
  C <- vector_sym(3)
  cumsum(C)
}

Define (invisibly) caracas symbols in global environment

Description

Define (invisibly) caracas symbols in global environment

Define symbol for components in vector

Usage

def_sym(..., charvec = NULL, warn = FALSE, env = parent.frame())

def_sym_vec(x, env = parent.frame())

Arguments

...

Names for new symbols, also supports non-standard evaluation
charvec

Take each element in this character vector and define as caracas symbols
warn

Warn if existing variable names are overwritten
env

The environment in which the assignment is made.
x

Character vector.

Value

Names of declared variables (invisibly)

See Also

symbol(), as_sym()

Examples

if (has_sympy()) {
  ls()
  def_sym(n1, n2, n3)
  ls()
  def_sym("x1", "x2", "x3")
  ls()
  # def_sym("x1", "x2", "x3", warn = TRUE) # Do not run as will cause a warning
  ls()
  def_sym(i, j, charvec = c("x", "y"))
  ls()
}

if (has_sympy()) {
  def_sym(z1, z2, z3)
  u <- paste0("u", seq_len(3))
  ## Creates symbols u1, u2, u3 and binds to names u1, u2, u3 in R.
  def_sym_vec(u) 
  ## Same as (but easier than)
  def_sym(u1, u2, u3)
  ## Notice: this creates matrix [u1, u2, u3]
  as_sym(u)
 }

Symbolic differentiation of an expression

Description

Symbolic differentiation of an expression

Usage

der(expr, vars, simplify = TRUE)

Arguments

expr

A caracas_symbol
vars

variables to take derivate with respect to
simplify

Simplify result

Examples

if (has_sympy()) {
  x <- symbol("x")
  y <- symbol("y")
  f <- 3*x^2 + x*y^2
  der(f, x)
  g <- der(f, list(x, y))
  g
  dim(g)
  G <- matrify(g)
  G
  dim(G)
  
  h <- der(g, list(x, y))
  h
  dim(h)
  as.character(h)
  H <- matrify(h)
  H
  dim(H)
  
  g %>% 
    der(list(x, y), simplify = FALSE) %>% 
    der(list(x, y), simplify = FALSE) %>% 
    der(list(x, y), simplify = FALSE)
}

Symbolic differentiation of second order of an expression

Description

Symbolic differentiation of second order of an expression

Usage

der2(expr, vars, simplify = TRUE)

Arguments

expr

A caracas_symbol
vars

variables to take derivate with respect to
simplify

Simplify result

Examples

if (has_sympy()) {
  x <- symbol("x")
  y <- symbol("y")
  f <- 3*x^2 + x*y^2
  der2(f, x)
  h <- der2(f, list(x, y))
  h
  dim(h)
  H <- matrify(h)
  H
  dim(H)
}

Matrix diagonal

Description

Matrix diagonal

Usage

diag(x, ...)

Arguments

x

Object x
...

Passed on

Symbolic diagonal matrix

Description

Symbolic diagonal matrix

Usage

diag_(x, n = 1L, declare_symbols = TRUE, ...)

Arguments

x

Character vector with diagonal
n

Number of times x should be repeated
declare_symbols

Passed on to as_sym() when constructing symbolic matrix
...

Passed on to rep(x, n, ...)

Examples

if (has_sympy()) {
  diag_(c(1,3,5))
  diag_(c("a", "b", "c"))
  diag_("a", 2)
  diag_(vector_sym(4))
}

Replace matrix diagonal

Description

Replace matrix diagonal

Usage

diag(x) <- value

Arguments

x

Object x
value

Replacement value

Matrix diagonal

Description

Matrix diagonal

Usage

## S3 method for class 'caracas_symbol'
diag(x, ...)

Arguments

x

Object x
...

Not used

Replace diagonal

Description

Replace diagonal

Usage

## S3 replacement method for class 'caracas_symbol'
diag(x) <- value

Arguments

x

A caracas_symbol.
value

Replacement value

Examples

if (has_sympy()) {
  A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
  B <- as_sym(A)
  B
  diag(B)
  diag(B) <- "b"
  B
  diag(B)
}

Difference matrix

Description

Difference matrix

Usage

diff_mat(N, l = "-1", d = 1)

Arguments

N

Number of rows (and columns)
l

Value / symbol below main diagonal
d

Value / symbol on main diagonal

Examples

if (has_sympy()){
  Dm <- diff_mat(4)
  Dm
  y <- vector_sym(4, "y")
  Dm %*% y
}

Dimensions of a caracas symbol

Description

Dimensions of a caracas symbol

Usage

## S3 method for class 'caracas_symbol'
dim(x)

Arguments

x

caracas symbol

Dimensions of a caracas symbol

Description

Dimensions of a caracas symbol

Usage

## S3 replacement method for class 'caracas_symbol'
dim(x) <- value

Arguments

x

caracas symbol
value

new dimension

Examples

if (has_sympy()) {
  v <- vector_sym(4)
  v
  dim(v)
  dim(v) <- c(2, 2)
  v
  m <- matrix_sym(2, 2)
  dim(m)
  dim(m) <- c(4, 1)
  m
}

Do linear algebra operation

Description

Do linear algebra operation

Usage

do_la(x, slot, ...)

Arguments

x

A matrix for which a property is requested
slot

The property requested
...

Auxillary arguments

Value

Returns the requested property of a matrix.

Examples

if (has_sympy()) {
  A <- matrix(c("a", "0", "0", "1"), 2, 2) %>% as_sym()
  
  do_la(A, "QR")
  QRdecomposition(A)

  do_la(A, "LU")
  LUdecomposition(A)

  do_la(A, "cholesky", hermitian = FALSE)
  chol(A, hermitian = FALSE)
  
  do_la(A, "singular_value_decomposition")
  do_la(A, "svd")
  svd_res <- svd_(A)
  svd_res
  U_expr <- svd_res$U |> as_expr()
  U_expr
  eval(U_expr, list(a = 3+2i))
  
  b <- symbol("b", real = TRUE)
  B <- matrix(c("b", "0", "0", "1"), 2, 2) %>% as_sym(declare_symbols = FALSE)
  svd_(B)

  do_la(A, "eigenval")
  eigenval(A)
  
  do_la(A, "eigenvec")
  eigenvec(A)
  
  do_la(A, "inv")
  inv(A)
  
  do_la(A, "trace")
  trace_(A)
  
  do_la(A, "echelon_form")
  do_la(A, "rank")
  
  do_la(A, "det") # Determinant
  det(A)
}

Perform calculations setup previously

Description

Perform calculations setup previously

Usage

doit(x)

Arguments

x

A caracas_symbol

Examples

if (has_sympy()) {
   x <- symbol('x')
   res <- lim(sin(x)/x, "x", 0, doit = FALSE)
   res 
   doit(res)
}

Remove remainder term

Description

Remove remainder term

Usage

drop_remainder(x)

Arguments

x

Expression to remove remainder term from

See Also

taylor()

Examples

if (has_sympy()) {
  def_sym(x)
  f <- cos(x)
  ft_with_O <- taylor(f, x0 = 0, n = 4+1)
  ft_with_O
  ft_with_O %>% drop_remainder() %>% as_expr()
}

Create a symbol from a string

Description

Create a symbol from a string

Usage

eval_to_symbol(x)

Arguments

x

String to evaluate

Value

A caracas_symbol

Examples

if (has_sympy()) {
   x <- symbol('x')
   (1+1)*x^2
   lim(sin(x)/x, "x", 0)
}

Expand expression

Description

Expand expression

Usage

expand(x, ...)

Arguments

x

A caracas_symbol
...

Pass on to SymPy's expand, e.g. force = TRUE

Examples

if (has_sympy()) {
  def_sym(x)
  y <- log(exp(x))
  simplify(y)
  expand(simplify(y))
  expand(simplify(y), force = TRUE)
  expand_log(simplify(y))
}

Expand a function expression

Description

Expand a function expression

Usage

expand_func(x)

Arguments

x

A caracas_symbol

Expand a logarithmic expression

Description

Note that force as described at https://docs.sympy.org/latest/tutorial/simplification.html#expand-log is used meaning that some assumptions are taken.

Usage

expand_log(x)

Arguments

x

A caracas_symbol

Examples

if (has_sympy()) {
  x <- symbol('x')
  y <- symbol('y')
  z <- log(x*y)
  z
  expand_log(z)
}

Expand a trigonometric expression

Description

Expand a trigonometric expression

Usage

expand_trig(x)

Arguments

x

A caracas_symbol

Expand expression

Description

Expand expression

Usage

factor_(x)

Arguments

x

A caracas_symbol

Examples

if (has_sympy()){
  def_sym(x, y, z)
  factor_(x**3 - x**2 + x - 1)
  factor_(x**2*z + 4*x*y*z + 4*y**2*z)
}

Get numerator and denominator of a fraction

Description

Get numerator and denominator of a fraction

Usage

fraction_parts(x)

numerator(x)

denominator(x)

Arguments

x

Fraction

Examples

if (has_sympy()) {
     x <- as_sym("a/b")
     frac <- fraction_parts(x)
     frac
     frac$numerator
     frac$denominator
 }

Get free symbol in expression

Description

Get free symbol in expression

Usage

free_symbols(x)

Arguments

x

Expression in which to get the free symbols in

Examples

if (has_sympy()) {
  def_sym(a, b)
  x <- (a - b)^4
  free_symbols(x)
}

Generate generic vectors and matrices

Description

Generate generic vectors and matrices.

Usage

vector_sym(n, entry = "v")

matrix_sym(nrow, ncol, entry = "v")

matrix_sym_diag(nrow, entry = "v")

matrix_sym_symmetric(nrow, entry = "v")

Arguments

n

Length of vector
entry

The symbolic name of each entry.
nrow, ncol

Number of rows and columns

Examples

if (has_sympy()) {
  vector_sym(4, "b")
  matrix_sym(3, 2, "a")
  matrix_sym_diag(4, "s")
  matrix_sym_symmetric(4, "s")
}

Get basis

Description

Get basis

Usage

get_basis(x)

Arguments

x

caracas vector / matrix

Examples

if (has_sympy()) {
  x <- vector_sym(3)
  get_basis(x)

  W <- matrix(c("r_1", "r_1", "r_2", "r_2", "0", "0", "u_1", "u_2"), nrow=4)
  W <- as_sym(W)
  get_basis(W)
}

Access 'py' object

Description

Get the 'py' object. Note that it gives you extra responsibilities when you choose to access the 'py' object directly.

Usage

get_py()

Value

The 'py' object with direct access to the library.

Examples

if (has_sympy()) {
  py <- get_py()
}

Access 'SymPy' directly

Description

Get the 'SymPy' object. Note that it gives you extra responsibilities when you choose to access the 'SymPy' object directly.

Usage

get_sympy()

Value

The 'SymPy' object with direct access to the library.

Examples

if (has_sympy()) {
  sympy <- get_sympy()
  sympy$solve("x**2-1", "x")
}

Check if 'SymPy' is available

Description

Check if 'SymPy' is available

Usage

has_sympy()

Value

TRUE if 'SymPy' is available, else FALSE

Examples

has_sympy()

Install 'SymPy'

Description

Install the 'SymPy' Python package into a virtual environment or Conda environment.

Usage

install_sympy(method = "auto", conda = "auto")

Arguments

method

Installation method. By default, "auto" automatically finds a method that will work in the local environment. Change the default to force a specific installation method. Note that the "virtualenv" method is not available on Windows.
conda

Path to conda executable (or "auto" to find conda using the PATH and other conventional install locations).

Value

None

Integrate a function

Description

If no limits are provided, the indefinite integral is calculated. Otherwise, if both limits are provided, the definite integral is calculated.

Usage

int(f, var, lower, upper, doit = TRUE)

Arguments

f

Function to integrate
var

Variable to integrate with respect to (either string or caracas_symbol)
lower

Lower limit
upper

Upper limit
doit

Evaluate the integral immediately (or later with doit())

Examples

if (has_sympy()) {
  x <- symbol("x")
  
  int(1/x, x, 1, 10)
  int(1/x, x, 1, 10, doit = FALSE)
  int(1/x, x)
  int(1/x, x, doit = FALSE)
  int(exp(-x^2/2), x, -Inf, Inf)
  int(exp(-x^2/2), x, -Inf, Inf, doit = FALSE)
}

Is object a caracas symbol

Description

Is object a caracas symbol

Usage

is_sym(x)

Arguments

x

object

Compute Jacobian

Description

Compute Jacobian

Usage

jacobian(expr, vars)

Arguments

expr

'caracas expression'.
vars

variables to take derivative with respect to

See Also

score(), hessian() der()

Examples

if (has_sympy()) {
  x <- paste0("x", seq_len(3))
  def_sym_vec(x)
  y1 <- x1 + x2
  y2 <- x1^2 + x3
  y <- c(y1, y2)
  jacobian(y, x)
  u <- 2 + 4*x1^2
  jacobian(u, x1)
}

Kronecker product of two matrices

Description

Computes the Kronecker product of two matrices.

Usage

## S4 method for signature 'caracas_symbol,caracas_symbol'
kronecker(X, Y, FUN = "*", make.dimnames = FALSE, ...)

Arguments

X, Y

matrices as caracas symbols.
FUN

a function; it may be a quoted string.
make.dimnames

Provide dimnames that are the product of the dimnames of ‘X’ and ‘Y’.
...

optional arguments to be passed to ‘FUN’.

Value

Kronecker product of A and B.

Examples

if (has_sympy()) {
  A <- matrix_sym(2, 2, "a")
  B <- matrix_sym(2, 2, "b")
  II <- matrix_sym_diag(2)
  EE <- eye_sym(2,2)
  JJ <- ones_sym(2,2)

  kronecker(A, B)
  kronecker(A, B, FUN = "+")
  kronecker(II, B)
  kronecker(EE, B)
  kronecker(JJ, B)
}

Limit of a function

Description

Limit of a function

Usage

lim(f, var, val, dir = NULL, doit = TRUE)

Arguments

f

Function to take limit of
var

Variable to take limit for (either string or caracas_symbol)
val

Value for var to approach
dir

Direction from where var should approach val: '+' or '-'
doit

Evaluate the limit immediately (or later with doit())

Examples

if (has_sympy()) {
  x <- symbol("x")
  lim(sin(x)/x, "x", 0)
  lim(1/x, "x", 0, dir = '+')
  lim(1/x, "x", 0, dir = '-')
}

Do linear algebra operation

Description

Performs various linear algebra operations like finding the inverse, the QR decomposition, the eigenvectors and the eigenvalues.

Usage

columnspace(x, matrix = TRUE)

nullspace(x, matrix = TRUE)

rowspace(x, matrix = TRUE)

singular_values(x)

inv(x, method = c("gauss", "lu", "cf", "yac"))

eigenval(x)

eigenvec(x)

GramSchmidt(x)

pinv(x)

rref(x)

QRdecomposition(x)

LUdecomposition(x)

## S3 method for class 'caracas_symbol'
chol(x, ...)

svd_(x, ...)

det(x, ...)

trace_(x)

Arguments

x

A matrix for which a property is requested.
matrix

When relevant should a matrix be returned.
method

The default works by Gaussian elimination. The alternatives are $LU$ decomposition (lu), the cofactor method (cf), and Ryacas (yac).
...

Auxillary arguments.

Value

Returns the requested property of a matrix.

See Also

do_la()

Examples

if (has_sympy()) {
  A <- matrix(c("a", "0", "0", "1"), 2, 2) |> as_sym()
  
  QRdecomposition(A)
  LUdecomposition(A)
  #chol(A) # error
  chol(A, hermitian = FALSE)
  eigenval(A)
  eigenvec(A)
  inv(A)
  det(A)
  rowspace(A)
  columnspace(A)
  nullspace(A)
  
  ## Matrix inversion:
  d <- 3
  m <- matrix_sym(d, d)
  print(system.time(inv(m)))                  ## Gauss elimination
  print(system.time(inv(m, method="cf")))     ## Cofactor 
  print(system.time(inv(m, method="lu")))     ## LU decomposition
  if (requireNamespace("Ryacas")){
    print(system.time(inv(m, method="yac")))  ## Use Ryacas
  }

  A <- matrix(c("a", "b", "c", "d"), 2, 2) %>% as_sym()
  evec <- eigenvec(A)
  evec
  evec1 <- evec[[1]]$eigvec
  evec1
  simplify(evec1)
  
  lapply(evec, function(l) simplify(l$eigvec))

  A <- as_sym("[[1, 2, 3], [4, 5, 6]]")
  pinv(A)
}

Convert object to list of elements

Description

Convert object to list of elements

Usage

listify(x)

Arguments

x

Object

Examples

if (has_sympy()) {
  x <- as_sym("Matrix([[b1*x1/(b2 + x1)], [b1*x2/(b2 + x2)], [b1*x3/(b2 + x3)]])")
  listify(x)
  
  xT <- t(x)
  listify(xT)
}

List defined symbols

Description

List defined symbols

Usage

ls_sym()

Matrix power

Description

Matrix power

Usage

mat_pow(x, pow = "1")

Arguments

x

A caracas_symbol, a matrix.
pow

Power to raise matrix x to

Examples

if (has_sympy() && sympy_version() >= "1.6") {
  M <- matrix_(c("1", "a", "a", 1), 2, 2)
  M
  mat_pow(M, 1/2)
}

Math functions

Description

If x is a matrix, the function is applied component-wise.

Usage

## S3 method for class 'caracas_symbol'
Math(x, ...)

Arguments

x

caracas_symbol.
...

further arguments passed to methods

Creates matrix from array symbol

Description

Creates matrix from array symbol

Usage

matrify(x)

Arguments

x

Array symbol to convert to matrix

Examples

if (has_sympy()) {
  x <- symbol("x")
  y <- symbol("y")
  f <- 3*x^2 + x*y^2
  matrify(f)
  h <- der2(f, list(x, y))
  h
  dim(h)
  H <- matrify(h)
  H
  dim(H)
}

Symbolic matrix

Description

Symbolic matrix

Usage

matrix_(..., declare_symbols = TRUE)

Arguments

...

Passed on to matrix()
declare_symbols

Passed on to as_sym() when constructing symbolic matrix

Examples

if (has_sympy()) {
  matrix_(1:9, nrow = 3)
  matrix_("a", 2, 2)
}

Matrix cross product

Description

Matrix cross product

Usage

crossprod_(x, y = NULL)

tcrossprod_(x, y = NULL)

Arguments

x, y

caracas matrices

Matrix multiplication

Description

Matrix multiplication

Matrix multiplication

Usage

x %*% y

## S3 method for class 'caracas_symbol'
x %*% y

Arguments

x

Object x
y

Object y

See Also

base::%*%()

base::%*%()

Numerical evaluation

Description

Numerical evaluation

Usage

N(x, digits = 15)

Arguments

x

caracas object
digits

Number of digits

Examples

if (has_sympy()) {
  n_2 <- as_sym("2")
  n_pi <- as_sym("pi", declare_symbols = FALSE)
  x <- sqrt(n_2) * n_pi
  x
  N(x)
  N(x, 5)
  N(x, 50)
  as.character(N(x, 50))
}

Math operators

Description

Math operators

Usage

## S3 method for class 'caracas_symbol'
Ops(e1, e2)

Arguments

e1

A caracas_symbol.
e2

A caracas_symbol.

Print scaled matrix

Description

Print scaled matrix

Usage

## S3 method for class 'caracas_scaled_matrix'
print(x, ...)

Arguments

x

A caracas_scaled_matrix
...

Passed to print.caracas_symbol()

Print solution

Description

Print solution

Usage

## S3 method for class 'caracas_solve_sys_sol'
print(
  x,
  simplify = getOption("caracas.print.sol.simplify", default = TRUE),
  ...
)

Arguments

x

A caracas_symbol
simplify

Print solution in a simple format
...

Passed to print.caracas_symbol()

Examples

if (has_sympy()) {
  x <- symbol('x')
  solve_sys(x^2, -1, x)
  
  y <- symbol("y")
  lhs <- cbind(3*x*y - y, x)
  rhs <- cbind(-5*x, y+4)
  sol <- solve_sys(lhs, rhs, list(x, y))
  sol
}

Print symbol

Description

Print symbol

Usage

## S3 method for class 'caracas_symbol'
print(
  x,
  prompt = getOption("caracas.prompt", default = "c: "),
  method = getOption("caracas.print.method", default = "utf8"),
  rowvec = getOption("caracas.print.rowvec", default = TRUE),
  ...
)

Arguments

x

A caracas_symbol
prompt

Which prompt/prefix to print (default: 'c: ')
method

What way to print (utf8, prettyascii, ascii, compactascii)
rowvec

FALSE to print column vectors as is
...

not used

Product of a function

Description

Product of a function

Usage

prod_(f, var, lower, upper, doit = TRUE)

Arguments

f

Function to take product of
var

Variable to take product for (either string or caracas_symbol)
lower

Lower limit
upper

Upper limit
doit

Evaluate the product immediately (or later with doit())

Examples

if (has_sympy()) {
  x <- symbol("x")
  p <- prod_(1/x, "x", 1, 10)
  p
  as_expr(p)
  prod(1/(1:10))
  n <- symbol("n")
  prod_(x, x, 1, n)
}

Rank of matrix

Description

Rank of matrix

Usage

rankMatrix_(x)

Arguments

x

Numeric or symbolic matrix

Examples

if (has_sympy()) {
  X <- matrix_(paste0("x_",c(1,1,1,1,2,2,2,2,3,4,3,4)), nrow=4)
  X
  rankMatrix_(X)
  colspan(X)
}

Elementwise reciprocal matrix

Description

Elementwise reciprocal matrix

Usage

reciprocal_matrix(x, numerator = 1)

Arguments

x

Object x
numerator

The numerator in the result.

Examples

if (has_sympy()) {
  s <- as_sym("[[r1, r2, r3], [u1, u2, u3]]")
  reciprocal_matrix(s, numerator = 7)
}

Form Row and Column Sums

Description

Form Row and Column Sums

Usage

rowSums_(x)

colSums_(x)

Arguments

x

Symbolic matrix

Examples

if (has_sympy()) {
  X <- matrix_(paste0("x_",c(1,1,1,1,2,2,2,2,3,4,3,4)), nrow=4)
  rowSums_(X)
  colSums_(X)
}

Create list of factors as in a product

Description

Create list of factors as in a product

Usage

scale_matrix(X, k = NULL, divide = TRUE)

Arguments

X

matrix
k

scalar to be factored out
divide

Should X be divided with k before constructing scaled matrix?

Examples

if (has_sympy()) {
  V <- matrix_sym(2, 2, "v")
  a <- symbol("a")
  
  K <- a*V
  scale_matrix(K, a)
  scale_matrix(V, a, divide = FALSE)

  Ks <- scale_matrix(V, a, divide = FALSE)
  Ks
  W <- matrix_sym(2, 2, "w")
  unscale_matrix(Ks) %*% W
  unscale_matrix(Ks) %*% W |> scale_matrix(a)
  Ksi <- unscale_matrix(Ks) |> inv() |> scale_matrix(a/det(unscale_matrix(Ks)))
  (Ksi |> unscale_matrix()) %*% (Ks |> unscale_matrix()) |> simplify()
  tex(Ksi)
}

Score and Hessian matrix

Description

Compute column vector of first derivatives and matrix of second derivatives of univariate function.

Usage

score(expr, vars, simplify = TRUE)

hessian(expr, vars, simplify = TRUE)

Arguments

expr

'caracas expression'.
vars

variables to take derivative with respect to.
simplify

Try to simplify result using simplify(); may be time consuming.

See Also

jacobian(), der()

Examples

if (has_sympy()) {
  def_sym(b0, b1, x, x0)
  f <- b0 / (1 + exp(b1*(x-x0)))
  S <- score(f, c(b0, b1))
  S
  H <- hessian(f, c(b0, b1))
  H
}

Simplify expression

Description

Simplify expression

Usage

simplify(x)

Arguments

x

A caracas_symbol

Solve a linear system of equations

Description

Find x in Ax = b. If b not supplied, the inverse of A is returned.

Usage

solve_lin(A, b)

Arguments

A

matrix
b

vector

Examples

if (has_sympy()) {
  A <- matrix_sym(2, 2, "a")
  b <- vector_sym(2, "b")
  # Inverse of A:
  solve_lin(A) %*% A |> simplify()
  # Find x in Ax = b
  x <- solve_lin(A, b)
  A %*% x |> simplify()
}

Solves a system of non-linear equations

Description

If called as solve_sys(lhs, vars) the roots are found. If called as solve_sys(lhs, rhs, vars) the solutions to lhs = rhs for vars are found.

Usage

solve_sys(lhs, rhs, vars)

Arguments

lhs

Equation (or equations as row vector/1xn matrix)
rhs

Equation (or equations as row vector/1xn matrix)
vars

vector of variable names or symbols

Value

A list with solutions (with class caracas_solve_sys_sol for compact printing), each element containing a named list of the variables' values.

Examples

if (has_sympy()) {
  x <- symbol('x')
  exp1 <- 2*x + 2
  exp2 <- x
  solve_sys(cbind(exp1), cbind(exp2), x)
  
  x <- symbol("x")
  y <- symbol("y")
  lhs <- cbind(3*x*y - y, x)
  rhs <- cbind(-5*x, y+4)
  sol <- solve_sys(lhs, rhs, list(x, y))
  sol
}

Solve lower or upper triangular system

Description

Solve lower or upper triangular system

Usage

solve_lower_triangular(a, b, ...)

solve_upper_triangular(a, b, ...)

Arguments

a

caracas_symbol
b

If provided, either a caracas_symbol (if not, as_sym() is called on the object)
...

Not used

Examples

if (has_sympy()) {
  A <- matrix_sym(3, 3)
  A[upper.tri(A)] <- 0
  solve_lower_triangular(A) |> simplify()
  A <- matrix_sym(3, 3)
  A[lower.tri(A)] <- 0
  solve_upper_triangular(A) |> simplify()
}

Solve a System of Linear Equations

Description

Solve a System of Linear Equations

Usage

## S3 method for class 'caracas_symbol'
solve(a, b, ...)

Arguments

a

caracas_symbol
b

If provided, either a caracas_symbol (if not, as_sym() is called on the object)
...

Not used

Examples

if (has_sympy()) {
  A <- matrix_sym(2, 2, "a")
  b <- vector_sym(2, "b")
  # Inverse of A:
  solve(A)
  inv(A)
  solve(A) %*% A |> simplify()
  # Find x in Ax = b
  x <- solve(A, b)
  A %*% x |> simplify()
  solve(A, c(2, 1)) |> simplify()
}

Special matrices: zeros_sym, ones_sym, eye_sym

Description

Special matrices: zeros_sym, ones_sym, eye_sym

Usage

zeros_sym(nrow, ncol)

ones_sym(nrow, ncol)

eye_sym(nrow, ncol)

Arguments

nrow, ncol

Number of rows and columns of output

See Also

diag_(), matrix_sym(), vector_sym()

Examples

if (has_sympy()){
  zeros_sym(3, 4)
  ones_sym(3, 4)
  eye_sym(3, 4)
}

Substitute symbol for value

Description

Substitute symbol for value

Usage

subs(sym, nms, vls)

Arguments

sym

Expression
nms

Names of symbols (see Details)
vls

Values that nms is substituted with (see Details)

Details

Two different ways to call this function is supported:

  1. Supplying nms as a named list and omitting vls. If two components have the same name, the behaviour is undefined.

  2. Supplying both nms and vls See Examples.

Examples

if (has_sympy()) {
   x <- symbol('x')
   e <- 2*x^2
   e
   subs(e, "x", "2")
   subs(e, x, 2)
   subs(e, list(x = 2))
   
   A <- matrix_sym(2, 2, "a")
   B <- matrix_sym(2, 2, "b")
   e <- A %*% A
   subs(e, A, B)
}

Sum of a function

Description

Sum of a function

Usage

sum_(f, var, lower, upper, doit = TRUE)

Arguments

f

Function to take sum of
var

Variable to take sum for (either string or caracas_symbol)
lower

Lower limit
upper

Upper limit
doit

Evaluate the sum immediately (or later with doit())

Examples

if (has_sympy()) {
  x <- symbol("x")
  s <- sum_(1/x, "x", 1, 10)
  as_expr(s)
  sum(1/(1:10))
  n <- symbol("n")
  simplify(sum_(x, x, 1, n))
}

Summation

Description

Summation

Usage

## S3 method for class 'caracas_symbol'
sum(..., na.rm = FALSE)

Arguments

...

Elements to sum
na.rm

Not used

Ask type of caracas symbol

Description

Ask type of caracas symbol

Usage

sym_class(x)

Arguments

x

An object, a caracas object is expected

Ask if type of caracas symbol is of a requested type

Description

Ask if type of caracas symbol is of a requested type

Usage

sym_inherits(x, what)

Arguments

x

An object, a caracas object is expected
what

Requested type (e.g. atomic, vector, list, matrix)

Create a symbol

Description

Find available assumptions at https://docs.sympy.org/latest/modules/core.html#module-sympy.core.assumptions.

Usage

symbol(x, ...)

Arguments

x

Name to turn into symbol
...

Assumptions like positive = TRUE

Value

A caracas_symbol

See Also

as_sym()

Examples

if (has_sympy()) {
  x <- symbol("x")
  2*x
  
  x <- symbol("x", positive = TRUE)
  ask(x, "positive")
}

Ask type of caracas symbol

Description

Ask type of caracas symbol

Usage

symbol_class(x)

Arguments

x

An object, a caracas object is expected

Check if object is a caracas matrix

Description

Check if object is a caracas matrix

Usage

symbol_is_matrix(x)

Arguments

x

An object

Examples

if (has_sympy() && sympy_version() >= "1.6") {
  x <- vector_sym(4)
  symbol_is_matrix(x)   ## TRUE
  x2 <- as.character(x) ## "Matrix([[v1], [v2], [v3], [v4]])"
  symbol_is_matrix(x2)  ## TRUE
  x3 <- as_character_matrix(x) ## R matrix
  symbol_is_matrix(x3)  ## FALSE 
}

Call a SymPy function directly on x

Description

Extend caracas by calling SymPy functions directly.

Usage

sympy_func(x, fun, ...)

Arguments

x

Object to call fun on
fun

Function to call
...

Passed on to fun

Examples

if (has_sympy()) {
  def_sym(x, a)
  p <- (x-a)^4
  p
  q <- p %>% sympy_func("expand")
  q
  q %>% sympy_func("factor")
  
  def_sym(x, y, z)
  expr <- x*y + x - 3 + 2*x^2 - z*x^2 + x^3
  expr
  expr %>% sympy_func("collect", x) 
  
  x <- symbol("x")
  y <- gamma(x+3)
  sympy_func(y, "expand_func")
  expand_func(y)
}

Get 'SymPy' version

Description

Get 'SymPy' version

Usage

sympy_version()

Value

The version of the 'SymPy' available

Examples

if (has_sympy()) {
  sympy_version()
}

Transpose of matrix

Description

Transpose of matrix

Usage

## S3 method for class 'caracas_symbol'
t(x)

Arguments

x

If caracas_symbol treat as such, else call base::t().

Taylor expansion

Description

Taylor expansion

Usage

taylor(f, x0 = 0, n = 6)

Arguments

f

Function to be expanded
x0

Point to expand around
n

Order of remainder term

See Also

drop_remainder()

Examples

if (has_sympy()) {
  def_sym(x)
  f <- cos(x)
  ft_with_O <- taylor(f, x0 = 0, n = 4+1)
  ft_with_O
  ft_with_O %>% drop_remainder() %>% as_expr()
}

Export object to TeX

Description

Export object to TeX

Usage

tex(x, zero_as_dot = FALSE, matstr = NULL, ...)

Arguments

x

A caracas_symbol
zero_as_dot

Print zero as dots
matstr

Replace ⁠\begin{matrix}⁠ with another environment, e.g. pmatrix. If vector of length two, the second element is an optional argument.
...

Other arguments passed along

Examples

if (has_sympy()) {
S <- matrix_sym_symmetric(3, "s")
S[1, 2] <- "1-x"
S
tex(S)
tex(S, matstr = "pmatrix")
tex(S, matstr = c("pmatrix", "r"))
}

Export scaled matrix to tex

Description

Export scaled matrix to tex

Usage

## S3 method for class 'caracas_scaled_matrix'
tex(x, ...)

Arguments

x

scaled matrix
...

Other arguments passed along

Dump latex representation of sympy object.

Description

Dump latex representation of sympy object and compile document into pdf.

Usage

texshow(x)

Arguments

x

An object that can be put in latex format with caracas' tex() function or a character string with tex code (in math mode).

Value

Nothing, but a .tex file and a .pdf file is generated.

Examples

if (has_sympy()) {
S <- matrix_sym_symmetric(3, "s")
S
## Not run: 
texshow(S)
texshow(paste0("S = ", tex(S)))

## End(Not run)
}

Coerce caracas object

Description

Coerce caracas object

Usage

to_list(x)

to_vector(x)

to_matrix(x)

Arguments

x

a caracas object is expected

Convert object to tuple

Description

Convert object to tuple

Usage

tuplify(x)

Arguments

x

Object

Examples

if (has_sympy()) {
  x <- as_sym("Matrix([[b1*x1/(b2 + x1)], [b1*x2/(b2 + x2)], [b1*x3/(b2 + x3)]])")
  tuplify(x)
}

Remove inner-most dimension

Description

Remove inner-most dimension

Usage

unbracket(x)

Arguments

x

Array symbol to collapse dimension from

Examples

if (has_sympy()) {
  x <- as_sym(paste0("x", 1:3))
  y <- as_sym("y")
  l <- list(x, y)
  l
  unbracket(l)
}

Extract unique elements

Description

Extract unique elements

Usage

## S3 method for class 'caracas_symbol'
unique(x, incomparables = FALSE, ...)

Arguments

x

A caracas vector or matrix
incomparables

Same meaning as for other unique methods
...

Additional arguments; currently not used.

Examples

if (has_sympy()){
v <- vector_sym(4)
v2 <- rep(v, each=2)
unique(v2)
}

Extract matrix from scaled matrix

Description

Extract matrix from scaled matrix

Usage

unscale_matrix(X)

Arguments

X

scaled matrix created with scale_matrix()

Examples

if (has_sympy()) {
  V <- matrix_sym(2, 2, "v")
  a <- symbol("a")
  Ks <- scale_matrix(V, a, divide = FALSE)
  Ks
  unscale_matrix(Ks)
  V %*% a
}

Creates symbol vector from list of caracas symbols

Description

Creates symbol vector from list of caracas symbols

Usage

vectorfy(x)

Arguments

x

Symbol to be coerced to vector