---
title: "03 - Objects in 'caracas'"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{03 - Objects in 'caracas'}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r, message=FALSE}
library(caracas)
```

```{r, include = FALSE}
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

```{r}
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
```{r}
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.
```{r}
a + 1
a <- a + 1
a / b
```

A text representation of a symbol can be found as:
```{r}
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`
```{r}
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
```{r}
u; v; w; x; y; z
```

A third way for creating a symbol with `as_sym`.
First notice:
```{r}
as_sym("l1")
# same as symbol("l1")
l2 <- as_sym("l2"); l2
# same as def_sym("l2")
```

More interestingly
```{r}
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:
```{r}
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:
```{r}
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:

```{r}
u %>% to_list()
u %>% to_vector()
m %>% to_list()
m %>% to_vector()
```

The corresponding text representations are:
```{r}
u %>% to_list() %>% as.character()
u %>% to_vector() %>% as.character()
m %>% to_list() %>% as.character()
m %>% to_vector() %>% as.character()
```

Likewise:
```{r}
m %>% to_matrix()
u %>% to_matrix()
```

## Indexing

Let 
```{r}
v <- m %>% to_vector()
l <- m %>% to_list()
V <- matrix_sym(2, 2)
```

```{r}

```




## Quick start

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

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

```{r}
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

```{r}
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)

```

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

```{r}
eigenval(Binv)
eigenvec(Binv)
```