Package 'Ryacas' reference manual

Title:	R Interface to the 'Yacas' Computer Algebra System
Description:	Interface to the 'yacas' computer algebra system (<http://www.yacas.org/>).
Authors:	Mikkel Meyer Andersen [aut, cre, cph], Rob Goedman [aut, cph], Gabor Grothendieck [aut, cph], Søren Højsgaard [aut, cph], Grzegorz Mazur [aut, cph], Ayal Pinkus [aut, cph], Nemanja Trifunovic [cph] (UTF-8 part of yacas (src/yacas/include/yacas/utf8*))
Maintainer:	Mikkel Meyer Andersen <[email protected]>
License:	GPL
Version:	1.1.5
Built:	2024-09-08 04:28:01 UTC
Source:	https://github.com/r-cas/ryacas

R interface to yacas computer algebra package

Description

Ryacas allows one to use the yacas computer algebra package entirely from within R.

Details

Please read the "Getting started" vignette.

Extract or replace parts of an object

Description

Extract or replace parts of an object

Usage

## S3 method for class 'yac_symbol'
x[i, j]
## S3 method for class 'yac_symbol'
x[i, j]

Arguments

`x`	A `yac_symbol`.
`i`	row indices specifying elements to extract or replace
`j`	column indices specifying elements to extract or replace

Extract parts of an object

Description

Extract parts of an object

Usage

## S3 method for class 'yac_symbol'
x[[i]]
## S3 method for class 'yac_symbol'
x[[i]]

Arguments

`x`	A `yac_symbol`.
`i`	indices specifying elements to extract

Extract or replace parts of an object

Description

Extract or replace parts of an object

Usage

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

Arguments

`x`	A `yac_symbol`.
`i`	row indices specifying elements to extract or replace
`j`	column indices specifying elements to extract or replace
`value`	the value to replace `x[i, j]` by

Matrix multiplication

Description

Matrix multiplication

Usage

x %*% y
x %*% y

Arguments

`x`	A `yac_symbol`
`y`	A `yac_symbol`

If x is a yacas command as string, convert to a character vector/matrix in R. If x is a yac_symbol (e.g. from ysym()), then convert it to a numeric object if there are no variables or a character type if there are variables.

Usage

as_r(x)
as_r(x)

Arguments

`x`	yacas list or list of lists to convert

Details

In yacas a vector is a list, and a matrix is a list of lists.

Convert R vector/matrix to yacas vector (list) or matrix (list of lists)

Description

Convert R vector/matrix to yacas vector (list) or matrix (list of lists)

Usage

as_y(x)
as_y(x)

Arguments

`x`	R vector to convert

Convert yac symbol to character

Description

Convert yac symbol to character

Usage

## S3 method for class 'yac_symbol'
as.character(x, ...)
## S3 method for class 'yac_symbol'
as.character(x, ...)

Arguments

`x`	A `yac_symbol`
`...`	not used

Combine R Objects by Columns

Description

Combine R Objects by Columns

Usage

## S3 method for class 'yac_symbol'
cbind(..., deparse.level = 1)
## S3 method for class 'yac_symbol'
cbind(..., deparse.level = 1)

Arguments

`...`	Objects to bind
`deparse.level`	Not used

Find the derivative of yac symbol

Description

Find the derivative of yac symbol

Usage

## S3 method for class 'yac_symbol'
deriv(expr, ...)
## S3 method for class 'yac_symbol'
deriv(expr, ...)

Arguments

`expr`	A `yac_symbol`
`...`	variables as character vector to take derivate with respect to

Matrix Determinant

Description

From base::det().

Usage

det(x, ...)
det(x, ...)

Arguments

`x`	If `yac_symbol` treat as such, else call `base::det()`.
`...`	further arguments passed to `base::det()`

Examples

(x <- matrix(1:4, ncol = 2))
det(x)
det(ysym(x))
(x <- matrix(1:4, ncol = 2))
det(x)
det(ysym(x))

Matrix diagonals

Description

From base::diag().

Usage

diag(x, ...)
diag(x, ...)

Arguments

`x`	If `yac_symbol` treat as such, else call `base::diag()`.
`...`	further arguments passed to `base::diag()`

Matrix diagonals

Description

From base::diag().

Usage

diag(x) <- value
diag(x) <- value

Arguments

`x`	If `yac_symbol` treat as such, else call `⁠base::diag<-()⁠`.
`value`	New value for `diag(x)`

Find the Hessian matrix of yac symbol

Description

Find the Hessian matrix of yac symbol

Usage

Hessian(expr, ...)
Hessian(expr, ...)

Arguments

`expr`	A `yac_symbol`
`...`	variables as character vector to take Hessian with respect to

Integration of Functions

Description

If f is a yac_symbol, yacas's Integrate() is used. Else, stats::integrate() is used.

Usage

integrate(f, ...)
integrate(f, ...)

Arguments

`f`	Function to integrate. See details.
`...`	See details.

Details

Additional arguments:

yac_symbol: var, lower, upper
Else (stats::integrate()): lower, upper

Find the Jacobian matrix of yac symbol

Description

Find the Jacobian matrix of yac symbol

Usage

Jacobian(expr, ...)
Jacobian(expr, ...)

Arguments

`expr`	A `yac_symbol`
`...`	variables as character vector to take Jacobian with respect to

Limits

Description

If first argument is a yac_symbol, yacas's Limit() is used.

Usage

lim(...)
lim(...)

Arguments

...

See details.

Details

Arguments:

yac_symbol: f, var, val, from_left, from_right

Lower and upper triangular part of a matrix

Description

Lower and upper triangular part of a matrix

Usage

lower.tri(x, diag = FALSE)
lower.tri(x, diag = FALSE)

Arguments

`x`	If `yac_symbol` treat as such, else call `base::lower.tri()`/`base::upper.tri()`.
`diag`	Whether diagonal is included.

Math functions

Description

Math functions

Usage

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

Arguments

`x`	`yac_symbol`.
`...`	further arguments passed to methods

Math operators

Description

Math operators

Usage

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

Arguments

`e1`	A `yac_symbol`.
`e2`	A `yac_symbol`.

Matrix Power

Description

Matrix Power

Usage

pow(x, n, ...)

## Default S3 method:
pow(x, n, ...)
pow(x, n, ...)

## Default S3 method:
pow(x, n, ...)

Arguments

`x`	If `yac_symbol` treat as such, else call `pow.default()`.
`n`	`n`th power of the square matrix.
`...`	further arguments passed to `pow.default()`

Examples

(x <- matrix(c(1, 2, 2, 3), ncol = 2))
pow(x, 2)
pow(ysym(x), 2)
(x <- matrix(c(1, 2, 2, 3), ncol = 2))
pow(x, 2)
pow(ysym(x), 2)

Product of Vector Elements

Description

Product of Vector Elements

Usage

## S3 method for class 'yac_symbol'
prod(expr, ..., na.rm = FALSE)
## S3 method for class 'yac_symbol'
prod(expr, ..., na.rm = FALSE)

Arguments

`expr`	Expression to be multiplied
`...`	Not used
`na.rm`	Not used

Combine R Objects by Rows

Description

Combine R Objects by Rows

Usage

## S3 method for class 'yac_symbol'
rbind(..., deparse.level = 1)
## S3 method for class 'yac_symbol'
rbind(..., deparse.level = 1)

Arguments

`...`	Objects to bind
`deparse.level`	Not used

Simplify expression

Description

Simplify expression

Usage

simplify(x, timeout = 2)
simplify(x, timeout = 2)

Arguments

`x`	A `yac_symbol`
`timeout`	timeout in seconds before simplification is aborted; only works when package unix is available

Solve a system of equations

Description

This generic function solves the equation $a x = b$ for $x$.

Usage

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

Arguments

`a`	A `yac_symbol`
`b`	A `yac_symbol` or a value, see details and examples.
`...`	See details and examples.

Details

When a is a matrix and b not provided, this finds the inverse of a. When a is a matrix and a vector b is provided, the linear system of equations is solved.

Note that solving non-linear equations:

solve(a, b): find roots of a for variable b, i.e. yacas Solve(a == 0, b)
solve(a, b, v): find solutions to a == b for variable v, i.e. yacas Solve(a == b, v)

This also works for a system of equations (when a is a vector)

Examples

A <- outer(0:3, 1:4, "-") + diag(2:5)
a <- 1:4
B <- ysym(A)
b <- ysym(a)
solve(A)
solve(B)
solve(A, a)
solve(B, b)

poly <- ysym("x^2 - x - 6")
solve(poly, "x")    # Solve(poly == 0, x)
solve(poly, 3, "x") # Solve(poly == 3, x)

A <- outer(0:3, 1:4, "-") + diag(2:5)
a <- 1:4
B <- ysym(A)
b <- ysym(a)
solve(A)
solve(B)
solve(A, a)
solve(B, b)

poly <- ysym("x^2 - x - 6")
solve(poly, "x")    # Solve(poly == 0, x)
solve(poly, 3, "x") # Solve(poly == 3, x)

Summation

Description

If only expr given: sum elements.

Usage

## S3 method for class 'yac_symbol'
sum(expr, var, lower, upper, ..., na.rm = FALSE)
## S3 method for class 'yac_symbol'
sum(expr, var, lower, upper, ..., na.rm = FALSE)

Arguments

`expr`	Expression to be summed
`var`	Variable to sum
`lower`	Lower limit
`upper`	Upper limit
`...`	Not used
`na.rm`	Not used

Details

Else: sums expr by letting var taking values from lower to upper (potentially Inf)

t

Description

Usage

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

Arguments

`x`	If `yac_symbol` treat as such, else call `base::t()`.

Export object to TeX

Description

Export object to TeX

Usage

tex(x)
tex(x)

Arguments

`x`	A `yac_symbol`

Matrix Trace

Description

The trace of a square matrix is the sum of the diagonal elements.

Usage

tr(x, ...)

## Default S3 method:
tr(x, ...)
tr(x, ...)

## Default S3 method:
tr(x, ...)

Arguments

`x`	If `yac_symbol` treat as such, else call `tr.default()`.
`...`	further arguments passed to `tr.default()`

Examples

(x <- matrix(1:4, ncol = 2))
tr(x)
tr(ysym(x))
(x <- matrix(1:4, ncol = 2))
tr(x)
tr(ysym(x))

Lower and upper triangular part of a matrix

Description

Lower and upper triangular part of a matrix

Usage

upper.tri(x, diag = FALSE)
upper.tri(x, diag = FALSE)

Arguments

`x`	If `yac_symbol` treat as such, else call `base::lower.tri()`/`base::upper.tri()`.
`diag`	Whether diagonal is included.

Vectorize

Description

Vectorize

Usage

vec(x, ...)

## Default S3 method:
vec(x, ...)
vec(x, ...)

## Default S3 method:
vec(x, ...)

Arguments

`x`	If `yac_symbol` treat as such, else call `base::as.vector()`.
`...`	further arguments passed to `base::as.vector()`

Examples

(x <- matrix(1:9, ncol = 3))
vec(x)
vec(ysym(x))
(x <- matrix(1:9, ncol = 3))
vec(x)
vec(ysym(x))

Half-Vectorize

Description

Half-Vectorize

Usage

vech(x, ...)

## Default S3 method:
vech(x, ...)
vech(x, ...)

## Default S3 method:
vech(x, ...)

Arguments

`x`	If `yac_symbol` treat as such, else call `vech.default()`.
`...`	further arguments passed to `vech.default()`

Examples

A <- mtcars[, c(1, 3, 4, 5, 6, 7)]
x <- cov(A)
vech(x)
vech(ysym(x))
A <- mtcars[, c(1, 3, 4, 5, 6, 7)]
x <- cov(A)
vech(x)
vech(ysym(x))

Give a variable a value

Description

Give a variable a value

Usage

with_value(x, var, val)
with_value(x, var, val)

Arguments

`x`	yac_symbol
`var`	Variable
`val`	Value

Evaluate a yacas expression

Description

Evaluate a yacas expression by replacing variables with values as for the given list.

Usage

y_eval(expr, ..., as.r = FALSE)
y_eval(expr, ..., as.r = FALSE)

Arguments

`expr`	a valid yacas expression
`...`	a list of assignements (see example)
`as.r`	if TRUE, then the expression is evaluated as R (if any variable to be substituted in the expression is a vector, then a vector is returned). If it is FALSE (default), a yacc expression is returned, replacing scalar variables.

Examples

# Evaluate as yacas object
eq <- ysym("2*y+x^2+2*x-3")
y_eval(eq, x=3, y=2)

# Evaluate as R expression:
y_eval(eq, x=3, y=2, as.r=TRUE)
# This allows to use vectors:
y_eval(eq, x=1:10, y=2, as.r=TRUE)
# and to plot functions:
curve(y_eval(eq, x=x, y=2, as.r=TRUE), xlim=c(0,10))
# Evaluate as yacas object
eq <- ysym("2*y+x^2+2*x-3")
y_eval(eq, x=3, y=2)

# Evaluate as R expression:
y_eval(eq, x=3, y=2, as.r=TRUE)
# This allows to use vectors:
y_eval(eq, x=1:10, y=2, as.r=TRUE)
# and to plot functions:
curve(y_eval(eq, x=x, y=2, as.r=TRUE), xlim=c(0,10))

Prepare simple `yacas` call

Description

Prepare simple yacas call

Usage

y_fn(x, fn, ...)
y_fn(x, fn, ...)

Arguments

`x`	parameter to function `fn`
`fn`	function with parameter `x`
`...`	additional arguments to `fn`

Examples

y_fn("x^2 - 1", "TeXForm")
yac_str(y_fn("x^2 - 1", "TeXForm"))

y_fn("x^2 - 1", "Factor")
yac_str(y_fn("x^2 - 1", "Factor"))

cmd <- "x^2 - 1 == 0" %>% y_fn("Solve", "x")
cmd
sol <- yac_str(cmd)
sol
yac_str(y_rmvars(sol))

y_fn("x^2 - 1", "TeXForm")
yac_str(y_fn("x^2 - 1", "TeXForm"))

y_fn("x^2 - 1", "Factor")
yac_str(y_fn("x^2 - 1", "Factor"))

cmd <- "x^2 - 1 == 0" %>% y_fn("Solve", "x")
cmd
sol <- yac_str(cmd)
sol
yac_str(y_rmvars(sol))

Pretty print yacas strings

Description

Pretty print yacas strings

Usage

y_print(x)
y_print(x)

Arguments

`x`	yacas string, e.g. a matrix

Examples

A <- diag(4)
Ayac <- as_y(A)
y_print(Ayac)

B <- A
B[2, 2] <- "-t"
Byac <- as_y(B)
Byac
y_print(Byac)

A <- diag(4)
Ayac <- as_y(A)
y_print(Ayac)

B <- A
B[2, 2] <- "-t"
Byac <- as_y(B)
Byac
y_print(Byac)

Remove/strip variable names

Description

This only builds a yacas command. You need to also call yac_str(), yac_expr() or similar. This is the reason that it does not call yacas: it depends on how you want it returned (string, expression).

Usage

y_rmvars(x)
y_rmvars(x)

Arguments

`x`	yacas command

Examples

cmd <- "{x == 2, y == 4}"
yac_str(cmd)
yac_str(y_rmvars(cmd))

cmd <- "{x == 2, y == 4}"
yac_str(cmd)
yac_str(y_rmvars(cmd))

Run yacas command

Description

Run yacas command

Usage

yac(x, rettype = c("str", "expr", "silent"))
yac(x, rettype = c("str", "expr", "silent"))

Arguments

`x`	yacas command
`rettype`	`str` for string/character, `expr` for expression, `silent` for silent

Examples

yac("D(x) x^2 + 4*x")
yac("D(x) x^2 + 4*x", rettype = "str")
yac("D(x) x^2 + 4*x", rettype = "expr")
yac("D(x) x^2 + 4*x", rettype = "silent")

yac("D(x) x^2 + 4*x")
yac("D(x) x^2 + 4*x", rettype = "str")
yac("D(x) x^2 + 4*x", rettype = "expr")
yac("D(x) x^2 + 4*x", rettype = "silent")

Assign yacas variable

Description

Assign yacas variable

Usage

yac_assign(value, x)
yac_assign(value, x)

Arguments

`value`	Expression
`x`	Variable name

yacas command line interface

Description

Interactive interface to the yacas

Usage

yac_cli(enable_history = TRUE)
yac_cli(enable_history = TRUE)

Arguments

enable_history

Use R history such that previous yacas commands can be used. Default is TRUE.

Details

The user types valid yacas input and presses return. Type 'quit' to return to R prompt.

Value

Output of yacas is returned.

Note

Note that command will use R history() and modify it by default. Yacas is given a limited amount of time to complete, otherwise ⁠\[1\] CommandLine(1) : User interrupted calculation⁠ is returned. E.g. Taylor(x,0,5) 1/(1+x) will work, but Taylor(x,0,12) 1/(1+x) is likely to take too long.

References

https://yacas.sourceforge.io/

Examples


## Not run: 
yac_cli()
 (x+y)^3-(x-y)^3
 Simplify(%)
 q

## End(Not run)

## Not run: 
yac_cli()
 (x+y)^3-(x-y)^3
 Simplify(%)
 q

## End(Not run)

Run yacas command returning R expression

Description

Run yacas command returning R expression

Usage

yac_expr(x)
yac_expr(x)

Arguments

`x`	yacas command

Examples

yac_expr("D(x) x^2 + 4*x")
yac_expr("Limit(x, 1) (x^2 - 1)/(x - 1)")
yac_expr("Sum(n, 1, Infinity, (1/2)^n)")
yac_expr("Fibonacci(10)")
yac_expr("Sum(n, 1, 10, Fibonacci(n))")

yac_expr("D(x) x^2 + 4*x")
yac_expr("Limit(x, 1) (x^2 - 1)/(x - 1)")
yac_expr("Sum(n, 1, Infinity, (1/2)^n)")
yac_expr("Fibonacci(10)")
yac_expr("Sum(n, 1, 10, Fibonacci(n))")

Run yacas command silently

Description

Run yacas command silently

Usage

yac_silent(x)
yac_silent(x)

Arguments

`x`	yacas command

Run yacas command returning string/character

Description

Run yacas command returning string/character

Usage

yac_str(x)
yac_str(x)

Arguments

`x`	yacas command

Examples

yac_str("D(x) x^2 + 4*x")
yac_str("Limit(x, 1) (x^2 - 1)/(x - 1)")
yac_str("Sum(n, 1, Infinity, (1/2)^n)")
yac_str("Fibonacci(10)")
yac_str("Sum(n, 1, 10, Fibonacci(n))")
yac_str("TeXForm(x^2 - 1)")

yac_str("D(x) x^2 + 4*x")
yac_str("Limit(x, 1) (x^2 - 1)/(x - 1)")
yac_str("Sum(n, 1, Infinity, (1/2)^n)")
yac_str("Fibonacci(10)")
yac_str("Sum(n, 1, 10, Fibonacci(n))")
yac_str("TeXForm(x^2 - 1)")

Make a yacas symbol

Description

This is an alias for ysym(). See description there.

Usage

yac_symbol(x)
yac_symbol(x)

Arguments

`x`	A vector or a matrix

Value

A yac_symbol

Make a yacas symbol

Description

Note that this results in multiple calls to yacas and the performance may be slower than manually using e.g. yac_str().

Usage

ysym(x)
ysym(x)

Arguments

`x`	A vector or a matrix

Value

A yac_symbol

List defined yac_symbols

Description

List defined yac_symbols

Usage

ysym_ls(print_details = FALSE)
ysym_ls(print_details = FALSE)

Arguments

print_details

print content of symbols

Package 'Ryacas'

Help Index

R interface to yacas computer algebra package

Description

Details

Extract or replace parts of an object

Description

Usage

Arguments

Extract parts of an object

Description

Usage

Arguments

Extract or replace parts of an object

Description

Usage

Arguments

Matrix multiplication

Description

Usage

Arguments

Convert yacas object to R

Description

Usage

Arguments

Details

Convert R vector/matrix to yacas vector (list) or matrix (list of lists)

Description

Usage

Arguments

Convert yac symbol to character

Description

Usage

Arguments

Combine R Objects by Columns

Description

Usage

Arguments

Find the derivative of yac symbol

Description

Usage

Arguments

Matrix Determinant

Description

Usage

Arguments

Examples

Matrix diagonals

Description

Usage

Arguments

Matrix diagonals

Description

Usage

Arguments

Find the Hessian matrix of yac symbol

Description

Usage

Arguments

Integration of Functions

Description

Usage

Arguments

Details

Find the Jacobian matrix of yac symbol

Description

Usage

Arguments

Limits

Description

Usage

Arguments

Details

Lower and upper triangular part of a matrix

Description

Usage

Arguments

Math functions

Description

Usage

Prepare simple `yacas` call