assignment 1~5: fixes

- assignment05/pascal.mlw: lowered the difficulty (one more invariant given)
- assignment02, 03: minor fixes & divide into sub-problems

src/assignments/assignment02/vec_and_mat.rs

@@ -0,0 +1,121 @@
+//! Vector and matrices.
+//!
+//! You will implement simple operations on vectors and matrices.
+
+use std::ops::Mul;
+
+/// 2x2 matrix of the following configuration:
+///
+/// a, b
+/// c, d
+#[derive(Debug, Clone, Copy)]
+struct Mat2 {
+    a: u64,
+    b: u64,
+    c: u64,
+    d: u64,
+}
+
+/// 2x1 matrix of the following configuration:
+///
+/// a
+/// b
+#[derive(Debug, Clone, Copy)]
+struct Vec2 {
+    a: u64,
+    b: u64,
+}
+
+impl Mat2 {
+    /// Creates an identity matrix.
+    fn new() -> Self {
+        Self {
+            a: 1,
+            b: 0,
+            c: 0,
+            d: 1,
+        }
+    }
+}
+
+impl Mul<Mat2> for Mat2 {
+    type Output = Mat2;
+
+    fn mul(self, rhs: Mat2) -> Self::Output {
+        todo!()
+    }
+}
+
+impl Mul<Vec2> for Mat2 {
+    type Output = Vec2;
+
+    /// Multiplies the matrix by the vector.
+    fn mul(self, rhs: Vec2) -> Self::Output {
+        todo!()
+    }
+}
+
+impl Mat2 {
+    /// Calculates the power of matrix.
+    fn power(self, power: u64) -> Mat2 {
+        todo!()
+    }
+}
+
+impl Vec2 {
+    /// Gets the upper value of vector.
+    fn get_upper(self) -> u64 {
+        todo!()
+    }
+}
+
+/// The matrix used for calculating Fibonacci numbers.
+const FIBONACCI_MAT: Mat2 = Mat2 {
+    a: 1,
+    b: 1,
+    c: 1,
+    d: 0,
+};
+
+/// The vector used for calculating Fibonacci numbers.
+const FIBONACCI_VEC: Vec2 = Vec2 { a: 1, b: 0 };
+
+/// Calculates the Fibonacci number. (We assume the absence of integer overflow.)
+///
+/// Consult <https://web.media.mit.edu/~holbrow/post/calculating-fibonacci-numbers-with-matrices-and-linear-algebra/> for matrix computation of Fibonacci numbers.
+pub fn fibonacci(n: u64) -> u64 {
+    (FIBONACCI_MAT.power(n) * FIBONACCI_VEC).get_upper()
+}
+
+/// 2x2 floating-point matrix of the following configuration:
+///
+/// a, b
+/// c, d
+#[derive(Debug, Clone, Copy)]
+pub struct FMat2 {
+    /// row 1, column 1
+    pub a: f64,
+    /// row 1, column 2
+    pub b: f64,
+    /// row 2, column 1
+    pub c: f64,
+    /// row 2, column 2
+    pub d: f64,
+}
+
+impl FMat2 {
+    /// Returns the inverse of the given matrix. (We assume the given matrix is always invertible.)
+    /// HINT: https://www.mathcentre.ac.uk/resources/uploaded/sigma-matrices7-2009-1.pdf
+    ///
+    /// # Example
+    ///
+    /// ```
+    /// assert_eq!(
+    ///     Mat2 { a: 1.0, b: 1.0, c: 2.0, d: 3.0 }.inverse(),
+    ///     Mat2 { a: 3.0, b: -1.0, c: -2.0, d: 1.0}
+    /// );
+    /// ```
+    pub fn inverse(self) -> Self {
+        todo!()
+    }
+}