z <- cos(x)^2 + sin(x)^2
z
#> [caracas]: 2 2
#> sin (x) + cos (x)
simplify(z)
#> [caracas]: 1
tex(z)
#> [1] "\\sin^{2}{\\left(x \\right)} + \\cos^{2}{\\left(x \\right)}"
z <- cos(x)*cos(y) - sin(x)*sin(y)
z
#> [caracas]: -sin(x)⋅sin(y) + cos(x)⋅cos(y)
simplify(z)
#> [caracas]: cos(x + y)
z <- cos(x + y)
z
#> [caracas]: cos(x + y)
expand(z)
#> [caracas]: cos(x + y)
expand_trig(z)
#> [caracas]: -sin(x)⋅sin(y) + cos(x)⋅cos(y)
x <- symbol('x')
y <- symbol('y')
z <- log(x*y)
z
#> [caracas]: log(x⋅y)
expand_log(z)
#> [caracas]: log(x) + log(y)
x <- symbol("x")
intf(1/x, x, 1, 10)
#> [caracas]: log(10)
i1 <- intf(1/x, x, 1, 10, doit = FALSE)
i1
#> [caracas]: 10
#> ⌠
#> ⎮ 1
#> ⎮ ─ dx
#> ⎮ x
#> ⌡
#> 1
tex(i1)
#> [1] "\\int\\limits_{1}^{10} \\frac{1}{x}\\, dx"
doit(i1)
#> [caracas]: log(10)
intf(1/x, x)
#> [caracas]: log(x)
i1 <- intf(1/x, x, doit = FALSE)
i1
#> [caracas]: ⌠
#> ⎮ 1
#> ⎮ ─ dx
#> ⎮ x
#> ⌡
tex(i1)
#> [1] "\\int \\frac{1}{x}\\, dx"
doit(i1)
#> [caracas]: log(x)
x <- symbol("x")
limf(sin(x)/x, "x", 0)
#> [caracas]: 1
limf(1/x, "x", 0, dir = '+')
#> [caracas]: ∞
limf(1/x, "x", 0, dir = '-')
#> [caracas]: -∞
We can also postpone evaluation:
Note that the function is called d()
and not deriv()
.
x <- symbol("x")
y <- symbol("y")
f <- 3*x^2 + x*y^2
f
#> [caracas]: 2 2
#> 3⋅x + x⋅y
as_r(f)
#> expression(3 * x^2 + x * y^2)
der(f, "x")
#> [caracas]: 2
#> 6⋅x + y
der(f, x)
#> [caracas]: 2
#> 6⋅x + y
der(f, c("x", "y"))
#> [caracas]: ⎡ 2 ⎤
#> ⎣6⋅x + y 2⋅x⋅y⎦
der(f, c(x, y))
#> [caracas]: ⎡ 2 ⎤
#> ⎣6⋅x + y 2⋅x⋅y⎦
f1 <- der(f, c(x, y))
f1
#> [caracas]: ⎡ 2 ⎤
#> ⎣6⋅x + y 2⋅x⋅y⎦
as.character(f1)
#> [1] "[6*x + y^2, 2*x*y]"
as_r(f1)
#> expression(cbind(6 * x + y^2, 2 * x * y))
eval(as_r(f1), list(x = 1, y = 2))
#> [,1] [,2]
#> [1,] 10 4
der(f1, c(x, y))
#> [caracas]: ⎡ 6 2⋅y⎤
#> ⎢ ⎥
#> ⎣2⋅y 2⋅x⎦
f2 <- der2(f, c(x, y))
f2
#> [caracas]: ⎡ 6 2⋅y⎤
#> ⎢ ⎥
#> ⎣2⋅y 2⋅x⎦
as_r(f2)
#> expression(rbind(cbind(6, 2 * y), cbind(2 * y, 2 * x)))
eval(as_r(f2), list(x = 1, y = 2))
#> [,1] [,2]
#> [1,] 6 4
#> [2,] 4 2
A <- matrix(c("x", 0, 0, "2*x"), 2, 2)
A
#> [,1] [,2]
#> [1,] "x" "0"
#> [2,] "0" "2*x"
B <- as_symbol(A)
B
#> [caracas]: ⎡x 0 ⎤
#> ⎢ ⎥
#> ⎣0 2⋅x⎦
2*B
#> [caracas]: ⎡2⋅x 0 ⎤
#> ⎢ ⎥
#> ⎣ 0 4⋅x⎦
B*B # Component-wise / Hadamard product
#> [caracas]: ⎡ 2 ⎤
#> ⎢x 0 ⎥
#> ⎢ ⎥
#> ⎢ 2⎥
#> ⎣0 4⋅x ⎦
dim(B)
#> [1] 2 2
sqrt(B)
#> [caracas]: ⎡√x 0 ⎤
#> ⎢ ⎥
#> ⎣0 √2⋅√x⎦
log(B)
#> [caracas]: ⎡log(x) zoo ⎤
#> ⎢ ⎥
#> ⎣ zoo log(2⋅x)⎦
sum(B)
#> [caracas]: 3⋅x
B %*% t(B)
#> [caracas]: ⎡ 2 ⎤
#> ⎢x 0 ⎥
#> ⎢ ⎥
#> ⎢ 2⎥
#> ⎣0 4⋅x ⎦
diag(B)
#> [caracas]: [x 2⋅x]
cbind(B, B)
#> [caracas]: ⎡x 0 x 0 ⎤
#> ⎢ ⎥
#> ⎣0 2⋅x 0 2⋅x⎦
rbind(B, B)
#> [caracas]: ⎡x 0 ⎤
#> ⎢ ⎥
#> ⎢0 2⋅x⎥
#> ⎢ ⎥
#> ⎢x 0 ⎥
#> ⎢ ⎥
#> ⎣0 2⋅x⎦
A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
B <- as_symbol(A)
eigen_val(B)
#> [[1]]
#> [[1]]$eigval
#> [caracas]: a
#>
#> [[1]]$eigmult
#> [1] 2
#>
#>
#> [[2]]
#> [[2]]$eigval
#> [caracas]: 0
#>
#> [[2]]$eigmult
#> [1] 1
eigen_vec(B)
#> [[1]]
#> [[1]]$eigval
#> [caracas]: 0
#>
#> [[1]]$eigmult
#> [1] 1
#>
#> [[1]]$eigvec
#> [caracas]: [-1 0 1]ᵀ
#>
#>
#> [[2]]
#> [[2]]$eigval
#> [caracas]: a
#>
#> [[2]]$eigmult
#> [1] 2
#>
#> [[2]]$eigvec
#> [caracas]: [1 0 0]ᵀ
eigen(eval(as_r(B), list(a = 2)))
#> eigen() decomposition
#> $values
#> [1] 2 2 0
#>
#> $vectors
#> [,1] [,2] [,3]
#> [1,] 1 -1.000000e+00 -0.7071068
#> [2,] 0 2.220446e-16 0.0000000
#> [3,] 0 2.220446e-16 0.7071068
inv()
/ solve_lin()
solve_sys()
Below find an example with maximising the multinomial likelihood.
p <- as_symbol(paste0("p", 1:3))
y <- as_symbol(paste0("y", 1:3))
a <- as_symbol("a")
l <- sum(y*log(p))
l
#> [caracas]: y₁⋅log(p₁) + y₂⋅log(p₂) + y₃⋅log(p₃)
L <- -l + a*(sum(p) - 1)
L
#> [caracas]: a⋅(p₁ + p₂ + p₃ - 1) - y₁⋅log(p₁) - y₂⋅log(p₂) - y₃⋅log(p₃)
tex(L)
#> [1] "a \\left(p_{1} + p_{2} + p_{3} - 1\\right) - y_{1} \\log{\\left(p_{1} \\right)} - y_{2} \\log{\\left(p_{2} \\right)} - y_{3} \\log{\\left(p_{3} \\right)}"
g <- der(L, c(p, a))
g
#> [caracas]: ⎡ y₁ y₂ y₃ ⎤
#> ⎢a - ── a - ── a - ── p₁ + p₂ + p₃ - 1⎥
#> ⎣ p₁ p₂ p₃ ⎦
sol <- solve_sys(g, c(p, a))
sol
#> Solution 1:
#> p1 = y₁
#> ────────────
#> y₁ + y₂ + y₃
#> p2 = y₂
#> ────────────
#> y₁ + y₂ + y₃
#> p3 = y₃
#> ────────────
#> y₁ + y₂ + y₃
#> a = y₁ + y₂ + y₃
sol[[1L]]$p1
#> [caracas]: y₁
#> ────────────
#> y₁ + y₂ + y₃
tex(sol[[1L]]$p1)
#> [1] "\\frac{y_{1}}{y_{1} + y_{2} + y_{3}}"
x <- symbol('x')
eq <- 2*x^2 - x
eq
#> [caracas]: 2
#> 2⋅x - x
subs(eq, x, "y")
#> [caracas]: 2
#> 2⋅y - y
p <- as_symbol(paste0("p", 1:3))
y <- as_symbol(paste0("y", 1:3))
a <- as_symbol("a")
l <- sum(y*log(p))
L <- -l + a*(sum(p) - 1)
g <- der(L, c(a, p))
sols <- solve_sys(g, c(a, p))
sol <- sols[[1L]]
sol
#> $a
#> [caracas]: y₁ + y₂ + y₃
#>
#> $p1
#> [caracas]: y₁
#> ────────────
#> y₁ + y₂ + y₃
#>
#> $p2
#> [caracas]: y₂
#> ────────────
#> y₁ + y₂ + y₃
#>
#> $p3
#> [caracas]: y₃
#> ────────────
#> y₁ + y₂ + y₃
H <- der2(L, c(p, a))
H
#> [caracas]: ⎡ y₁ ⎤
#> ⎢─── 0 0 1⎥
#> ⎢ 2 ⎥
#> ⎢p₁ ⎥
#> ⎢ ⎥
#> ⎢ y₂ ⎥
#> ⎢ 0 ─── 0 1⎥
#> ⎢ 2 ⎥
#> ⎢ p₂ ⎥
#> ⎢ ⎥
#> ⎢ y₃ ⎥
#> ⎢ 0 0 ─── 1⎥
#> ⎢ 2 ⎥
#> ⎢ p₃ ⎥
#> ⎢ ⎥
#> ⎣ 1 1 1 0⎦
H_sol <- subs_lst(H, sol)
H_sol
#> [caracas]: ⎡ 2 ⎤
#> ⎢(y₁ + y₂ + y₃) ⎥
#> ⎢─────────────── 0 0 1⎥
#> ⎢ y₁ ⎥
#> ⎢ ⎥
#> ⎢ 2 ⎥
#> ⎢ (y₁ + y₂ + y₃) ⎥
#> ⎢ 0 ─────────────── 0 1⎥
#> ⎢ y₂ ⎥
#> ⎢ ⎥
#> ⎢ 2 ⎥
#> ⎢ (y₁ + y₂ + y₃) ⎥
#> ⎢ 0 0 ─────────────── 1⎥
#> ⎢ y₃ ⎥
#> ⎢ ⎥
#> ⎣ 1 1 1 0⎦
Note that all vectors in caracas
are column vectors.
A <- matrix(c("a", 0, 0, 0, "a", "a", "a", 0, 0), 3, 3)
B <- as_symbol(A)
B[, 2]
#> [caracas]: [0 a a]ᵀ
B[, -2]
#> [caracas]: ⎡a a⎤
#> ⎢ ⎥
#> ⎢0 0⎥
#> ⎢ ⎥
#> ⎣0 0⎦
B[1, ]
#> [caracas]: [a 0 a]ᵀ
B[1, , drop = FALSE] # Note this is a 1x3 matrix
#> [caracas]: [a 0 a]
B[, 2] <- "x"
B
#> [caracas]: ⎡a x a⎤
#> ⎢ ⎥
#> ⎢0 x 0⎥
#> ⎢ ⎥
#> ⎣0 x 0⎦
SymPy
directly# Multinomial likelihood
p <- as_symbol(paste0("p", 1:3))
y <- as_symbol(paste0("y", 1:3))
a <- as_symbol("a")
l <- sum(y*log(p))
L <- -l + a*(sum(p) - 1)
L
#> [caracas]: a⋅(p₁ + p₂ + p₃ - 1) - y₁⋅log(p₁) - y₂⋅log(p₂) - y₃⋅log(p₃)
print(L, ascii = TRUE)
#> [caracas]: a*(p1 + p2 + p3 - 1) - y1*log(p1) - y2*log(p2) - y3*log(p3)
g <- der(L, c(p, a))
sol <- solve_sys(g, c(p, a))
sol
#> Solution 1:
#> p1 = y₁
#> ────────────
#> y₁ + y₂ + y₃
#> p2 = y₂
#> ────────────
#> y₁ + y₂ + y₃
#> p3 = y₃
#> ────────────
#> y₁ + y₂ + y₃
#> a = y₁ + y₂ + y₃
print(sol, simplify = FALSE)
#> [[1]]
#> [[1]]$p1
#> [caracas]: y₁
#> ────────────
#> y₁ + y₂ + y₃
#>
#> [[1]]$p2
#> [caracas]: y₂
#> ────────────
#> y₁ + y₂ + y₃
#>
#> [[1]]$p3
#> [caracas]: y₃
#> ────────────
#> y₁ + y₂ + y₃
#>
#> [[1]]$a
#> [caracas]: y₁ + y₂ + y₃
as.character(g)
#> [1] "[a - y1/p1, a - y2/p2, a - y3/p3, p1 + p2 + p3 - 1]"
as_character_matrix(g)
#> [,1] [,2] [,3] [,4]
#> [1,] "a - y1/p1" " a - y2/p2" " a - y3/p3" " p1 + p2 + p3 - 1"
The following options are available:
caracas.print.prettyascii
caracas.print.ascii
caracas.print.rowvec
caracas.print.sol.simplify
sol
#> Solution 1:
#> p1 = y₁
#> ────────────
#> y₁ + y₂ + y₃
#> p2 = y₂
#> ────────────
#> y₁ + y₂ + y₃
#> p3 = y₃
#> ────────────
#> y₁ + y₂ + y₃
#> a = y₁ + y₂ + y₃
L
#> [caracas]: a⋅(p₁ + p₂ + p₃ - 1) - y₁⋅log(p₁) - y₂⋅log(p₂) - y₃⋅log(p₃)
options(caracas.print.prettyascii = TRUE)
sol
#> Solution 1:
#> p1 = y1
#> ------------
#> y1 + y2 + y3
#> p2 = y2
#> ------------
#> y1 + y2 + y3
#> p3 = y3
#> ------------
#> y1 + y2 + y3
#> a = y1 + y2 + y3
L
#> [caracas]: a*(p1 + p2 + p3 - 1) - y1*log(p1) - y2*log(p2) - y3*log(p3)
options(caracas.print.prettyascii = NULL) # reset to default (FALSE)
sol
#> Solution 1:
#> p1 = y₁
#> ────────────
#> y₁ + y₂ + y₃
#> p2 = y₂
#> ────────────
#> y₁ + y₂ + y₃
#> p3 = y₃
#> ────────────
#> y₁ + y₂ + y₃
#> a = y₁ + y₂ + y₃
L
#> [caracas]: a⋅(p₁ + p₂ + p₃ - 1) - y₁⋅log(p₁) - y₂⋅log(p₂) - y₃⋅log(p₃)
options(caracas.print.ascii = TRUE)
sol
#> Solution 1:
#> p1 = y1/(y1 + y2 + y3)
#> p2 = y2/(y1 + y2 + y3)
#> p3 = y3/(y1 + y2 + y3)
#> a = y1 + y2 + y3
L
#> [caracas]: a*(p1 + p2 + p3 - 1) - y1*log(p1) - y2*log(p2) - y3*log(p3)
options(caracas.print.ascii = NULL) # reset to default (FALSE)
p
#> [caracas]: [p₁ p₂ p₃]ᵀ
options(caracas.print.rowvec = FALSE)
p
#> [caracas]: ⎡p₁⎤
#> ⎢ ⎥
#> ⎢p₂⎥
#> ⎢ ⎥
#> ⎣p₃⎦
options(caracas.print.rowvec = NULL) # reset to default (TRUE)
sol
#> Solution 1:
#> p1 = y₁
#> ────────────
#> y₁ + y₂ + y₃
#> p2 = y₂
#> ────────────
#> y₁ + y₂ + y₃
#> p3 = y₃
#> ────────────
#> y₁ + y₂ + y₃
#> a = y₁ + y₂ + y₃
options(caracas.print.sol.simplify = FALSE)
sol
#> [[1]]
#> [[1]]$p1
#> [caracas]: y₁
#> ────────────
#> y₁ + y₂ + y₃
#>
#> [[1]]$p2
#> [caracas]: y₂
#> ────────────
#> y₁ + y₂ + y₃
#>
#> [[1]]$p3
#> [caracas]: y₃
#> ────────────
#> y₁ + y₂ + y₃
#>
#> [[1]]$a
#> [caracas]: y₁ + y₂ + y₃
options(caracas.print.sol.simplify = NULL) # reset to default (TRUE)