# 03 - Objects in 'caracas In caracas: Computer Algebra

knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )  library(caracas)  inline_code <- function(x) { x } if (!has_sympy()) { # SymPy not available, so the chunks shall not be evaluated knitr::opts_chunk$set(eval = FALSE)

inline_code <- function(x) {
deparse(substitute(x))
}
}


## Variables, names, symbols etc.

We can think of a variable as a piece of memory in a computer. A variable typically also has a name (also called a symbol). That name/symbol is used to refer to the variable; that is, the name / symbol is a handle on the variable. It is like the difference between you and your name.

There are different ways of creating a variable in caracas. One is as

symbol("a")


which creates a SymPy variable a but provides no handle on it (no R-symbol). We can get an R-handle on a SymPy variable with

b <- symbol("a")
a <- symbol("b")


where we do something very confusing: Assign the R-name a to the SymPy variable b and vice versa. We can compute on variable b in SymPy by manipulating the symbol a in R, e.g.

a + 1
a <- a + 1
a / b


A text representation of a symbol can be found as:

a %>% print.default()
a %>% as.character()


Usually, the best practice is to assign R symbols to SymPy variables of the same name. To avoid confusion, symbol names and Python variable names will always coincide.

## Creating symbols

In addition to symbol() illustrated above, multiple R-symbols / Python-variables can be defined using def_sym and def_sym_vec

def_sym(u, v)
def_sym("w", "x")
def_sym_vec(c("y", "z"))


With this, R-symbols u, v, w, x exist and each are connected to Python variables with the same name

u; v; w; x; y; z


A third way for creating a symbol with as_sym. First notice:

as_sym("l1")
# same as symbol("l1")
l2 <- as_sym("l2"); l2
# same as def_sym("l2")


More interestingly

m_ <- paste0("m", 1:4)
m <- as_sym(m_)
m

B_ <- matrix(c("x", 2, 0, "2*x"), 2, 2)
B <- as_sym(B_)


## Classes

Above, r is a $4 \times 1$ matrix, while e.g. u is an atom:

m %>% symbol_class()
u %>% symbol_class()


We can coerce between different "classes" (we quote the word because it is not a class system as e.g. those known from R) A text representation of the variables are:

m %>% as.character()
u %>% as.character()


While not often needed that are also lists and vectors in Python. In caracas they are created by coercion:

u %>% to_list()
u %>% to_vector()
m %>% to_list()
m %>% to_vector()


The corresponding text representations are:

u %>% to_list() %>% as.character()
u %>% to_vector() %>% as.character()
m %>% to_list() %>% as.character()
m %>% to_vector() %>% as.character()


Likewise:

m %>% to_matrix()
u %>% to_matrix()


## Indexing

Let

v <- m %>% to_vector()
l <- m %>% to_list()
V <- matrix_sym(2, 2)




## Quick start

def_sym('x', 'y')
eq <- 2*x^2 - x - y
eq
as.character(eq)
as_expr(eq)
tex(eq)


$$r inline_code(tex(eq))$$

sol <- solve_sys(eq, x)
sol
# Access solutions
sol[[1]]$x sol[[2]]$x

dx <- der(eq, x)
dx
dx %>% symbol_class()

dxy <- der(eq, c(x, y))
dxy
dxy %>% symbol_class()

subs(eq, x, y)


## Linear algebra

B_ <- matrix(c("x", 2, 0, "2*x"), 2, 2)
B <- as_sym(B_)
B
Binv <- inv(B) # or solve_lin(B)
Binv
tex(Binv)
det(B)
Binv * det(B)


$$r inline_code(tex(Binv))$$

eigenval(Binv)
eigenvec(Binv)


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caracas documentation built on Oct. 17, 2023, 5:08 p.m.