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    <metadata lang="rus">
      <title>Метод Рунге-Кутта пятого порядка с автоматическим выбором шага</title>
      <author>Кирилл Николаев, Андрей Ивашов</author>
      <description>Решение обыкновенных дифференциальных уравнений методом Рунге-Кутта пятого порядка с автоматическим выбором шага.
Пользователь определяет коэффициенты исходного дифференциального уравнения, задачу Коши, границы отрезка и точность работы метода.
Программа преобразовывает коэффициенты уравнения к системе и проводит вычисления по методу Рунге-Кутта пятого порядка. При этом автоматически определяется оптимальный шаг работы алгоритма с учётом требуемой точности.
После вычислений с помощью функций интерполяции кубическими сплайнами строятся графики численного решения.</description>
      <company>SMath, http://smath.info/</company>
      <keywords>Примеры, Метод Рунге-Кутта, Дифференциальные уравнения, Графики</keywords>
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    <metadata lang="ind">
      <title>Metode Runge–Kutta orde lima dengan langkah adaptif</title>
      <author>Kirill Nikolaev, Andrey Ivashov</author>
      <translator>Waluyo Adi Siswanto</translator>
      <description>Penyelesain persamaan diferensial biasa menggunakan metode Runge-Kutta dengan langkah adaptif.
Pengguna perlu menyatakan koefisien dari persamaan asal, angka awal Cauchy (persoalan bilangan awal atau initial value problem), panjang langkah serta keakuratan perhitungan.
Program akan merubah persmaan asal menjadi satu sistem persamaan simultan kemudian memulai evaluasi dengan metode Runge–Kutta orde lima.
Algoritma secara otomatis memilih langkah optimal iterasi agar memperoleh keakuratan yang diinginkan.
Setelah selesai perhitungan, program menggambarkan grafik penyelasaian numerik dari hasil interpolasi kubik data.</description>
      <company>SMath, http://smath.info/</company>
      <keywords>Contoh, Metode Runge-Kutta, Persamaan diferensial, Grafik</keywords>
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    <metadata lang="eng">
      <title>Fifth-order Runge–Kutta method with adaptive step</title>
      <author>Kirill Nikolaev, Andrey Ivashov</author>
      <translator>Andrey Ivashov</translator>
      <description>Solution of ordinary differential equations using Fifth-order Runge–Kutta method with adaptive step.
User defines initial equation coefficients, a Cauchy problem (initial value problem), segment limits and calculations precision.
Program converts equation to the system of equations and starts evaluation with the Fifth-order Runge–Kutta method.
Algorithm automatically choose the optimal step of the iterations in respect to the specified accuracy.
After calculations program represents the graphs of numeric solution using cubic splines interpolation.</description>
      <company>SMath, http://smath.info/</company>
      <keywords>Examples, Runge-Kutta Method, Differential equations, Graphs</keywords>
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    <text lang="rus">
      <p bold="true">Преобразование уравнения к системе уравнений</p>
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    <text lang="ind">
      <p bold="true">Merubah satu persamaan diferensial menjadi satu sistem persamaan</p>
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      <p bold="true">Converting the equation to the system of equations</p>
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      <p>Исходное уравнение</p>
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    <text lang="ind">
      <p>Persamaan asal</p>
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    <text lang="eng">
      <p>Source Equation</p>
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      <p>Входные данные:</p>
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    <text lang="ind">
      <p>Data masukan:</p>
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    <text lang="eng">
      <p>Input data:</p>
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      <p>Коэффициенты уранения</p>
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    <text lang="ind">
      <p>Koefisien persamaan asal</p>
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    <text lang="eng">
      <p>Equation coefficients</p>
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      <p>Расчёт:</p>
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    <text lang="ind">
      <p>Perhitungan:</p>
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    <text lang="eng">
      <p>Calculation:</p>
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      <p>Степень уравнения</p>
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    <text lang="ind">
      <p>Orde persamaan asal</p>
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    <text lang="eng">
      <p>Equation power</p>
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    <text lang="rus">
      <p>Ответ:</p>
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    <text lang="ind">
      <p>Hasil:</p>
    </text>
    <text lang="eng">
      <p>Result:</p>
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        <e type="operand">5</e>
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        <e type="bracket">(</e>
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        <e type="operator" args="1">-</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="4">mat</e>
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  <region id="13" left="243" top="459" width="323" height="24" color="#000000" bgColor="#ffff80" fontSize="10">
    <text lang="rus">
      <p>Правая часть полученной системы</p>
    </text>
    <text lang="ind">
      <p>Suku kanan dari sistem persamaan baru</p>
    </text>
    <text lang="eng">
      <p>Right part of the result Equations set</p>
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  <region id="14" left="9" top="513" width="310" height="48" color="#0000ff" bgColor="#ffffff" fontSize="12">
    <text lang="rus">
      <p bold="true">Метод Рунге-Кутта пятого порядка
с автоматическим выбором шага</p>
    </text>
    <text lang="ind">
      <p bold="true">Metode Runge-Kutta orde lima
dengan langkah adaptif</p>
    </text>
    <text lang="eng">
      <p bold="true">Fifth-order Runge–Kutta method
with adaptive step</p>
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  </region>
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    <text lang="rus">
      <p>Входные данные:</p>
    </text>
    <text lang="ind">
      <p>Data masukan:</p>
    </text>
    <text lang="eng">
      <p>Input data:</p>
    </text>
  </region>
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        <e type="operand">t</e>
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        <e type="operand">f</e>
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    <text lang="rus">
      <p>Задача Коши</p>
    </text>
    <text lang="ind">
      <p>ProAngka awal Cauchy</p>
    </text>
    <text lang="eng">
      <p>A Cauchy problem</p>
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  <region id="20" left="90" top="666" width="102" height="24" color="#000000" bgColor="#ffff80" fontSize="10">
    <text lang="rus">
      <p>Начальная точка</p>
    </text>
    <text lang="ind">
      <p>Titik permulaan</p>
    </text>
    <text lang="eng">
      <p>Start Point</p>
    </text>
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    <math>
      <input>
        <e type="operand">T</e>
        <e type="operand">6</e>
        <e type="operator" args="2">:</e>
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  <region id="22" left="90" top="693" width="127" height="24" color="#000000" bgColor="#ffff80" fontSize="10">
    <text lang="rus">
      <p>Длина отрезка</p>
    </text>
    <text lang="ind">
      <p>Panjang langkah</p>
    </text>
    <text lang="eng">
      <p>Segment length</p>
    </text>
  </region>
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      <input>
        <e type="operand">ε</e>
        <e type="operand">10</e>
        <e type="operand">2</e>
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        <e type="operator" args="2">:</e>
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  <region id="24" left="90" top="729" width="86" height="24" color="#000000" bgColor="#ffff80" fontSize="10">
    <text lang="rus">
      <p>Точность</p>
    </text>
    <text lang="ind">
      <p>Keakuratan</p>
    </text>
    <text lang="eng">
      <p>Precision</p>
    </text>
  </region>
  <region id="25" left="9" top="765" width="111" height="24" border="true" color="#000000" bgColor="#dddddd" fontSize="10">
    <text lang="rus">
      <p>Расчёт:</p>
    </text>
    <text lang="ind">
      <p>Perhitungan:</p>
    </text>
    <text lang="eng">
      <p>Calculation:</p>
    </text>
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    <math>
      <input>
        <e type="operand">h</e>
        <e type="operand">0.1</e>
        <e type="operator" args="2">:</e>
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        <e type="operator" args="2">+</e>
        <e type="operand">tx</e>
        <e type="operand">k1</e>
        <e type="operand">k2</e>
        <e type="operator" args="2">+</e>
        <e type="operand">2</e>
        <e type="operator" args="2">/</e>
        <e type="operator" args="2">+</e>
        <e type="function" args="2">f</e>
        <e type="function" preserve="true" args="1">eval</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">:</e>
        <e type="operand">k4</e>
        <e type="operand">h</e>
        <e type="operand">3</e>
        <e type="operator" args="2">/</e>
        <e type="operand">j</e>
        <e type="operand">h</e>
        <e type="operand">2</e>
        <e type="operator" args="2">/</e>
        <e type="operator" args="2">+</e>
        <e type="operand">tx</e>
        <e type="operand">3</e>
        <e type="operand">k1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">9</e>
        <e type="operand">k3</e>
        <e type="operator" args="2">*</e>
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        <e type="operand">8</e>
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        <e type="operator" args="2">:</e>
        <e type="operand">k5</e>
        <e type="operand">h</e>
        <e type="operand">3</e>
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        <e type="operand">j</e>
        <e type="operand">h</e>
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        <e type="operand">3</e>
        <e type="operand">k1</e>
        <e type="operator" args="2">*</e>
        <e type="operand">9</e>
        <e type="operand">k3</e>
        <e type="operator" args="2">*</e>
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        <e type="operand">2</e>
        <e type="operator" args="2">/</e>
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        <e type="operand">6</e>
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        <e type="function" preserve="true" args="1">eval</e>
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        <e type="operand">δ</e>
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        <e type="operand">9</e>
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        <e type="operator" args="2">*</e>
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        <e type="operand">1</e>
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        <e type="operator" args="2">-</e>
        <e type="operator" args="2">:</e>
        <e type="operand">δ</e>
        <e type="function" preserve="true" args="1">min</e>
        <e type="function" preserve="true" args="1">abs</e>
        <e type="operand">5</e>
        <e type="operand">ε</e>
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        <e type="operator" args="2">≤</e>
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        <e type="operator" args="2">&amp;</e>
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        <e type="operand">k4</e>
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        <e type="function" preserve="true" args="2">augment</e>
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        <e type="operand">x</e>
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        <e type="function" preserve="true" args="3">el</e>
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        <e type="operand">h</e>
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        <e type="operand">h</e>
        <e type="operand">2</e>
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        <e type="operand">2</e>
        <e type="operand">1</e>
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        <e type="operand">8</e>
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  </region>
  <region id="30" left="180" top="1305" width="142" height="24" color="#000000" bgColor="#ffff80" fontSize="10">
    <text lang="rus">
      <p>Число значений</p>
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    <text lang="ind">
      <p>Angka hasil</p>
    </text>
    <text lang="eng">
      <p>Number of values</p>
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    <text lang="rus">
      <p>Анализ полученных данных:</p>
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    <text lang="ind">
      <p>Analisis data:</p>
    </text>
    <text lang="eng">
      <p>Data analysis:</p>
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        <e type="operand">x</e>
        <e type="function" preserve="true" args="3">cinterp</e>
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    <math>
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        <e type="operand">X</e>
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        <e type="function" preserve="true" args="1">transpose</e>
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    </math>
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    <math>
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        <e type="operand">x</e>
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    </plot>
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