数学 - JavaScript - 線型代数 - 行列の積(3 x 3、一般化、数学的帰納法)

学習環境/開発環境

数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
OS X El Capitan - Apple (OS)
Emacs (Text Editor)
JavaScript (プログラミング言語)
kjs-math-matrix (JavaScript Library)
Safari(Web browser)

線型代数入門 (松坂和夫(著)、岩波書店)の第3章(線型写像)、7(行列の積)、問6.を取り組んでみる。

問6.

\begin{array}{l} A = (\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) \\ A^{2} = (\begin{matrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}) \\ A^{3} = (\begin{matrix} 1 & 3 & 6 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{matrix}) \\ A^{4} = (\begin{matrix} 1 & 4 & 10 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{matrix}) \\ A^{5} = (\begin{matrix} 1 & 5 & 15 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \end{matrix}) \\ A^{n} = (\begin{matrix} 1 & n & \frac{n (n + 1)}{2} \\ 0 & 1 & n \\ 0 & 0 & 1 \end{matrix}) \\ n = k \\ A^{k} = (\begin{matrix} 1 & k & \frac{k (k + 1)}{2} \\ 0 & 1 & k \\ 0 & 0 & 1 \end{matrix}) \\ A^{k + 1} = A^{k} (\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) \\ = (\begin{matrix} 1 & k + 1 & \frac{(k + 1) (k + 2)}{2} \\ 0 & 1 & k + 1 \\ 0 & 0 & 1 \end{matrix}) \end{array}

JavaScript

コード(Emacs)


<div id="answer0"></div>

<script src="matrix.js"></script>
<script src="test.js"></script>


  // matrix.js
var Matrix;

Matrix = function (m, n, fill) {
    var fill = fill || 0,
        matrix = [],
        i,
        j,
        row;

    for (i = 0; i < m; i += 1) {
        row = [];
        for (j = 0; j < n; j += 1) {
            row.push(fill);
        }
        matrix.push(row);
    }
    this.rowLength = m;
    this.colomnLength = n;
    this.getElement = function (i, j) {
        this.isValidIndex(m, n, i, j);
        return matrix[i-1][j-1];
    };
    this.setElement = function (i, j, elem) {
        this.isValidIndex(m, n, i, j);
        matrix[i-1][j-1] = elem;
    };
};
Matrix.prototype.isValidIndex = function (m, n, i, j) {
    if (i < 1 || m < i || j < 1 || n < j) {
        throw {
            type: 'Index error',
            message: m + ' x ' + n + ': ' + '(' + i + ', ' + j + ')'
        };
    }
};
Matrix.prototype.toArray = function () {
    var matrix = [],
        i,
        j,
        m = this.rowLength,
        n = this.colomnLength,
        row;

    for (i = 1; i <= m; i += 1) {
        row = [];
        for (j = 1; j <= n; j += 1) {
            row.push(this.getElement(i, j))
        }
        matrix.push(row);
    }
    return matrix;
};
Matrix.prototype.toString = function () {
    var m = this.rowLength,
        n = this.colomnLength,
        i,
        j,
        result = '<math><mfenced><mtable>';

    if (typeof window !== 'undefined') {
        for (i = 1; i <= m; i += 1) {
            result += '<mtr>';
            for (j = 1; j <= n; j += 1) {
                result += '<mtd><mn>' + this.getElement(i, j) + '</mn></mtd>';
            }
            result += '</mtr>';
        }
        result += '</mtable></math>';
        return result;
    }
    return this.toArray();
};
Matrix.prototype.getRow = function (i) {
    var j,
        n = this.colomnLength,
        row = new Matrix(1, n);

    for (j = 1; j <= n; j += 1) {
        row.setElement(1, j, this.getElement(i, j));
    }
    return row;
};
Matrix.prototype.getColomn = function (j) {
    var m = this.rowLength,
        i,
        col = new Matrix(m, 1);

    for (i = 1; i <= m; i += 1) {
        col.setElement(i, 1, this.getElement(i, j));
    }
    return col;
};
Matrix.prototype.setRow = function (i, row) {
    var n = this.colomnLength,
        j;
    
    for (j = 1; j <= n; j += 1) {
        this.setElement(i, j, row.getElement(1, j));
    }
};
Matrix.prototype.setColomn = function (j, colomn) {
    var m = this.rowLength,
        i;
    
    for (i = 1; i <= m; i += 1) {
        this.setElement(i, j, colomn.getElement(i, 1));
    }
};
Matrix.prototype.isEqual = function (mat) {
    var m = this.rowLength,
        n = this.colomnLength,
        i,
        j;

    if (m !== mat.rowLength || n !== mat.colomnLength) {
        return false;
    }
    for (i = 1; i <= m; i += 1) {
        for (j = 1; j <= n; j += 1) {
            if (this.getElement(i, j) !== mat.getElement(i, j)) {
                return false;
            }
        }
    }
    return true;
};
Matrix.prototype.add = function (mat) {
    var m = this.rowLength,
        n = this.colomnLength,
        matrix = new Matrix(m, n),
        i,
        j;

    for (i = 1; i <= m; i += 1) {
        for (j = 1; j <= n; j += 1) {
            matrix.setElement(i, j,
                              this.getElement(i, j) + mat.getElement(i, j));
        }
    }
    return matrix;
};
Matrix.prototype.mulMatrix = function (mat) {
    var m = this.rowLength,
        n = this.colomnLength,
        m0 = mat.rowLength,
        n0 = mat.colomnLength,
        i,
        j,
        matrix = new Matrix(m, n0),
        elem,
        j0;

    if (n !== m0) {
        throw {
            type: "mulMatrix error",
            message: '(' + m + ', ' + n + ') x (' + m0 + ', ' + n0 + ')',
        };
    }
    for (i = 1; i <= m; i += 1) {
        for (j = 1; j <= n0; j += 1) {
            elem = 0;
            for (j0 = 1; j0 <= n; j0 += 1) {
                elem += this.getElement(i, j0) * mat.getElement(j0, j);
            }
            matrix.setElement(i, j, elem);
        }
    }
    return matrix;
};
Matrix.prototype.mulScalar = function (scalar) {
    var m = this.rowLength,
        n = this.colomnLength,
        i,
        j,
        matrix = new Matrix(m, n);

    for (i = 1; i <= m; i += 1) {
        for (j = 1; j <= n; j += 1) {
            matrix.setElement(i, j, this.getElement(i, j) * scalar);
        }
    }
    return matrix;
};

Array.prototype.toRow = function () {
    var len = this.length,
        matrix = new Matrix(1, len),
        j;

    for (j = 1; j <= len; j += 1) {
        matrix.setElement(1, j, this[j - 1]);
    }
    return matrix;
};
Array.prototype.toColomn = function () {
    var len = this.length,
        matrix = new Matrix(len, 1),
        i;

    for (i = 1; i <= len; i += 1) {
        matrix.setElement(i, 1, this[i - 1]);
    }
    return matrix;
};
Array.prototype.toMatrix = function () {
    var m = this.length,
        n = this[0].length,
        matrix = new Matrix(m, n),
        i,
        j;

    for (i = 1; i <= m; i += 1) {
        for (j = 1; j <= n; j += 1) {
            matrix.setElement(i, j, this[i-1][j-1]);
        }
    }
    return matrix;
};

// test
var a = [[1, 1, 1],
         [0, 1, 1],
         [0, 0, 1]].toMatrix(),
    b = a,
    i,
    n = 5,
    result = '',
    answer0 = document.querySelector('#answer0');

for (i = 0; i < n; i += 1) {
    result += b + ' ';
    b = b.mulMatrix(a);
}

answer0.innerHTML = result;

Kamimura's blog

2016年7月31日日曜日

数学 - JavaScript - 線型代数 - 行列の積(3 x 3、一般化、数学的帰納法)

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