# 99 - Reference 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))
}
}

## Quick start

x <- symbol('x')
as.character(x)
x
as_expr(x)
2*x
y <- symbol('y')
sqrt(3*x^y)
z <- cos(x)^2 + sin(x)^2
z
simplify(z)
tex(z)
z <- cos(x)*cos(y) - sin(x)*sin(y)
z
simplify(z)
z <- cos(x + y)
z
expand(z)
expand_trig(z)
x <- symbol('x')
y <- symbol('y')
z <- log(x*y)
z
expand_log(z)

### Sums

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

### Products

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)

### Integrals

x <- symbol("x")

int(1/x, x, 1, 10)
i1 <- int(1/x, x, 1, 10, doit = FALSE)
i1
tex(i1)
doit(i1)
int(1/x, x)
i1 <- int(1/x, x, doit = FALSE)
i1
tex(i1)
doit(i1)

### Limits

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

We can also postpone evaluation:

x <- symbol("x")
lim(sin(x)/x, "x", 0)
lim(sin(x)/x, x, 0)
res <- lim(sin(x)/x, "x", 0, doit = FALSE)
res
as.character(res)
tex(res)
doit(res)
as_expr(res)

### Derivatives

Note that the function is called d() and not deriv().

x <- symbol("x")
y <- symbol("y")
f <- 3*x^2 + x*y^2
f
as_expr(f)
der(f, "x")
der(f, x)
der(f, c("x", "y"))
der(f, list(x, y))
f1 <- der(f, list(x, y))
f1
as.character(f1)
as_expr(f1)
eval(as_expr(f1), list(x = 1, y = 2))
der(f1, list(x, y))
f2 <- der2(f, list(x, y))
f2
as_expr(f2)
eval(as_expr(f2), list(x = 1, y = 2))
x <- symbol("x")
y <- symbol("y")
f <- eval_to_symbol("[3*x**2 + x*y**2, 2*x, 5*y]")
f
der(f, list(x, y))

### Taylor expansion

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()

## Linear algebra

A <- matrix(c("x", 0, 0, "2*x"), 2, 2)
A
B <- as_sym(A)
B
2*B
B*B # Component-wise / Hadamard product
dim(B)
sqrt(B)
log(B)
sum(B)
B %*% t(B)
diag(B)
cbind(B, B)
rbind(B, B)
det(B)
QRdecomposition(B)
A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
B <- as_sym(A)
eigenval(B)
eigenvec(B)
eigen(eval(as_expr(B), list(a = 2)))
B
diag(B)
diag(B) <- "b"
B
diag(B)[-2] <- "a"
B

## Solve

• Linear system of equations: inv() / solve_lin()
• Non-linear system of equations: solve_sys()

Below find an example with maximising the multinomial likelihood.

p <- as_sym(paste0("p", 1:3))
y <- as_sym(paste0("y", 1:3))
a <- as_sym("a")
l <- sum(y*log(p))
l
L <- -l + a*(sum(p) - 1)
L
tex(L)
g <- der(L, list(p, a))
g
sol <- solve_sys(g, list(p, a))
sol
sol[[1L]]\$p1
tex(sol[[1L]]\$p1)

### Assumptions

x <- symbol("x", positive = TRUE)
solve_sys(x^2 - 1, 0, x)

x <- symbol("x", real = TRUE)
solve_sys(x^2 + 1, 0, x)

x <- symbol("x")
solve_sys(x^2 + 1, 0, x)

## Substitution

x <- symbol('x')
eq <- 2*x^2 - x
eq
subs(eq, x, "y")
p <- as_sym(paste0("p", 1:3))
y <- as_sym(paste0("y", 1:3))
a <- as_sym("a")
l <- sum(y*log(p))
L <- -l + a*(sum(p) - 1)
g <- der(L, c(a, p))
sols <- solve_sys(g, list(a, p))
sol <- sols[[1L]]
sol
H <- der2(L, list(p, a))
H
H_sol <- subs(H, sol)
H_sol

## Subsetting

Note that all vectors in caracas are column vectors.

A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
B <- as_sym(A)
B[, 2]
B[, -2]
B[1, ]
B[1, , drop = FALSE] # Note this is a 1x3 matrix
B[, 2] <- "x"
B

## Interactively show \$\LaTeX\$ representation

texshow(B)

plots the following in the plot window:

texshow(B)

You can also provide a string instead:

texshow(paste0("B = ", tex(B)))

giving

texshow(paste0("B = ", tex(B)))

## Using SymPy directly

sympy <- get_sympy()
sympy\$diff("2*a*x", "x")
sympy\$solve("x**2 - 1", "x")

## Assumptions

Below we give a brief example of assumptions. First consider the Cholesky decomposition of a matrix:

A <- matrix(c("x+1", 1, 1, 1), 2, 2) %>% as_sym()
A
do_la(A, "cholesky")

This fails as A is not positive (semi-)definite.

To ensure this, we need to impose restrictions on x. This is done by defining a symbol with an assumption about positivity:

y <- symbol("y", positive = TRUE)

We continue and define B, where it is important that declare_symbols = FALSE or else a new y will automatically be defined by caracas overwriting the above definition:

B <- as_sym("[[y + 1, 1], [1, 1]]", declare_symbols = FALSE)
B
do_la(B, "cholesky")

## Output

# Multinomial likelihood
p <- as_sym(paste0("p", 1:3))
y <- as_sym(paste0("y", 1:3))
a <- as_sym("a")
l <- sum(y*log(p))
L <- -l + a*(sum(p) - 1)
L
print(L, ascii = TRUE)
g <- der(L, list(p, a))
sol <- solve_sys(g, list(p, a))
sol
print(sol, simplify = FALSE)
as.character(g)
as_character_matrix(g)

### Options

The following options are available:

• caracas.print.method (utf8 is default, others are: prettyascii, ascii, compactascii)
• caracas.print.rowvec
• caracas.print.sol.simplify
sol
L
options(caracas.print.method = "prettyascii")
sol
L
options(caracas.print.method = "ascii")
sol
L
options(caracas.print.method = NULL) # Or 'utf8'
sol
L
p
options(caracas.print.rowvec = FALSE)
p
options(caracas.print.rowvec = NULL) # reset to default (TRUE)
sol
options(caracas.print.sol.simplify = FALSE)
sol
options(caracas.print.sol.simplify = NULL) # reset to default (TRUE)

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