上／下三角、對稱矩陣

December 12, 2021
上三角矩陣是對角線以下的元素均為 0，也就是 Aᵢⱼ = 0，i > j，例如：
1 2 3 4 5
0 6 7 8 9
0 0 10 11 12
0 0 0 13 14
0 0 0 0 15
下三角矩陣是對角線以上的元素均為 0，也就是 Aᵢⱼ = 0，i < j，例如：
1 0 0 0 0
2 6 0 0 0
3 7 10 0 0
4 8 11 13 0
5 9 12 14 15
對稱矩陣是元素對稱於對角線，例如：
1 2 3 4 5
2 6 7 8 9
3 7 10 11 12
4 8 11 13 14
5 9 12 14 15

上三角或下三角矩陣大部份元素不儲存值（為 0），可以使用一維陣列來儲存以節省空間，對稱矩陣因為對稱於對角線，可以使用上三角或下三角矩陣來儲存。

解法思路

假設矩陣為 n x n，為了計算方便，讓陣列索引由 1 開始，上三角矩陣化為一維陣列，以列為主的公式為：

loc = n * (i - 1) - i * (i - 1)/ 2 + j

以行為主的公式為：

loc = j * (j - 1) / 2 + i

下三角矩陣化為一維陣列，以列為主的公式為：

loc = i * (i - 1) / 2 + j

以行為主的公式為：

loc = n * (j - 1) - j * (j - 1) / 2 + i

公式的導證是由等差級數公式得到，對於索引由 0 開始的語言來說，只要將 i 與 j 各加 1，求得 loc 後減 1，就可套用以上的公式。

程式實作

#include <stdio.h> 
#include <stdlib.h> 
#define N 5 


int main(void) { 
    int arr1[N][N] = { 
        {1, 2, 3,  4,   5}, 
        {0, 6, 7,  8,   9}, 
        {0, 0, 10, 11, 12}, 
        {0, 0, 0,  13, 14}, 
        {0, 0, 0,  0,  15}}; 

    int arr2[N*(1+N)/2] = {0}; 

    int i, j, loc = 0; 

    printf("\n以列為主："); 
    for(i = 0; i < N; i++) { 
        for(j = 0; j < N; j++) { 
            if(arr1[i][j] != 0) 
                arr2[loc++] = arr1[i][j]; 
        } 
    } 
    for(i = 0; i < N*(1+N)/2; i++) 
        printf("%d ", arr2[i]); 

    printf("\n輸入索引(i, j)："); 
    scanf("%d, %d", &i, &j); 
    loc = N*i - i*(i+1)/2 + j; 
    printf("(%d, %d) = %d", i, j, arr2[loc]); 

    printf("\n"); 

    return 0; 
} 

import java.util.*;

public class TriangleArray {
    private List<Integer> list;
    private int length;
    
    public TriangleArray(int[][] array) {
        length = array.length;
        list = new ArrayList<Integer>(length * (1 + length) / 2);
        for(int i = 0; i < length; i++) { 
            for(int j = 0; j < length; j++) { 
                if(array[i][j] != 0) 
                    list.add(array[i][j]);
            } 
        }
    }
    
    public int get(int i, int j) {
        return list.get(length * i - i * (i + 1) / 2 + j);
    }

    public static void main(String[] args) {
        int[][] array = {{1, 2, 3,  4,   5}, 
                         {0, 6, 7,  8,   9}, 
                         {0, 0, 10, 11, 12}, 
                         {0, 0, 0,  13, 14}, 
                         {0, 0, 0,  0,  15}};
        
        TriangleArray triangleArray = new TriangleArray(array);
        System.out.print(triangleArray.get(2, 2));
    }
}

class TriangleArray:
    def __init__(self, array):
        self.__length = len(array)
        self.__list = []
        for i in range(self.__length):
            for j in range(self.__length):
                if array[i][j] != 0:
                    self.__list.append(array[i][j])
    def get(self, i, j):
        return self.__list[self.__length * i - i * (i + 1) // 2 + j]

array = [ 
            [1, 2, 3,  4,   5], 
            [0, 6, 7,  8,   9], 
            [0, 0, 10, 11, 12], 
            [0, 0, 0,  13, 14], 
            [0, 0, 0,  0,  15]
        ]

triangleArray = TriangleArray(array)
print(triangleArray.get(2, 2))

class TriangleArray(array: Array[Array[Int]]) {
    val length = array.length;
    val arr = (for(i <- 0 until length; j <- 0 until length
                   if array(i)(j) != 0
               ) yield array(i)(j)).toArray
               
    def get(i: Int, j: Int) = {
        arr(length * i - i * (i + 1) / 2 + j)
    }
}

val array = Array(
               Array(1, 2, 3,  4,   5), 
               Array(0, 6, 7,  8,   9), 
               Array(0, 0, 10, 11, 12), 
               Array(0, 0, 0,  13, 14), 
               Array(0, 0, 0,  0,  15)
            )
val triangleArray = new TriangleArray(array)
print(triangleArray.get(2, 2))

class TriangleArray
    def initialize(array)
        @length = array.length
        @list = []
        @length.times { |i|
            @length.times { |j|
                if array[i][j] != 0
                    @list << array[i][j]
                end
            }
        }
    end
    def get(i, j)
        @list[@length * i - i * (i + 1) / 2 + j]
    end
end


array = [ 
            [1, 2, 3,  4,   5], 
            [0, 6, 7,  8,   9], 
            [0, 0, 10, 11, 12], 
            [0, 0, 0,  13, 14], 
            [0, 0, 0,  0,  15]
        ]

triangleArray = TriangleArray.new(array)
puts triangleArray.get(2, 2)

class TriangleArray {
    constructor(array) {
        this.__length = array.length;
        this.__list = [];
        for(let i = 0; i < this.__length; i++) {
            for(let j = 0; j < this.__length; j++) {
                if(array[i][j] != 0) {
                    this.__list.push(array[i][j]);
                }
            }
        }
    }
    
    elem(i, j) {
        return this.__list[Math.floor(this.__length * i - i * (i + 1) / 2) + j];
    }
}

let array = [ 
    [1, 2, 3,  4,   5], 
    [0, 6, 7,  8,   9], 
    [0, 0, 10, 11, 12], 
    [0, 0, 0,  13, 14], 
    [0, 0, 0,  0,  15]
];

triangleArray = new TriangleArray(array);
console.log(triangleArray.elem(2, 2));

data TArray a = TArray Int [a]

triangleArray xs =
    let row = length xs
    in TArray row [xs !! i !! j | i <- [0..row - 1], j <- [0..(length $ xs !! 0) - 1], (xs !! i !! j) /= 0]

elemOf i j (TArray row xs) =
    xs !! (row * i - (i * (i + 1)) `div` 2 + j)
    
main = 
    let 
        triArray = triangleArray [ 
            [1, 2, 3,  4,   5], 
            [0, 6, 7,  8,   9], 
            [0, 0, 10, 11, 12], 
            [0, 0, 0,  13, 14], 
            [0, 0, 0,  0,  15]]
    in
    do
        print $ elemOf 2 2 triArray

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