# SMath

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## SMath Studio in the Cloud

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Create new worksheet form scratch or upload already existing document to put it into SMath Cloud. You can start editing worksheet immediately after it is created and continue working from any other place using generated link (using your account).

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Open worksheet you have previously created inside SMath Cloud. To use this feature you should be logged in.

### Examples (18)

Try examples created by members of SMath community to see different features provided by application. Opened documents can be easily edited in order to test them or use with your own input data.

Numeric integration method (Simpson's rule)
Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. User specifies function to integrate, interval and the number of iterations. At the end of calculation program controls the result with a built-in numerical integration function.
Legendre polynomials solving
Solving of Legendre polynomials defined by Rodrigues' formula. User specifies a power of the polinomial to get it's roots. Additionally represented graphs of first five Legendre functions.
Planetary gear with internal teeth
Animation in SMath Studio shown by the example of planetary gear with internal teeth.
Nonlinear systems of equations solving with Newton's method
Newton's method of the nonlinear systems of equations solving. This algorithm can be used to solve standalone equation as well. User specifies system of the equations, first approximations of the roots and the result accuracy. While calculation Jacobi matrix is created. Number of steps (iterations) of the While loop also displayed for the analysis purposes.
Jacobi matrix and Jacobian
Algorithm of Jacobi matrix generation and the definition of the Jacobian. The user specifies a function to construct Jacobian matrix in the loop using partial derivatives. The last step defines the functions to work with the result. All calculations are performs in symbols, with the possibility to get Symbolic and Numeric results of the algorithm.
Nonlinear equations solving with chord method
Nonlinear equations solving with chord method. User defines initial equation to proceed, calculation precision and the range. Program returns root of the initial equation, result accuracy and number of iterations.
Arabic to Roman numeral conversion
Algorithm for conversion Arabic numerals to Roman numerals. User specifies a number using Arabic digits. Program shows a result of conversion in Roman numerals form.
Hermite polynomials solving
Solving of Hermite polynomials. User specifies a power of the polinomial to get it's roots. Additionally represented graphs of first five Hermite functions.
Beam load calculation bearing with two supports
Calculation of the Beam load bearing with two supports to find stresses values of the supports. Worksheet requires to specify any number of the Point and/or Uniform Loads. Every input and output data supports values with Units.
Expansion of function to Maclaurin series
Expansion of the user-defined function to Maclaurin series with a custom degree.
Euclidean algorithm (calculating the GCD)
Efficient method for computing the greatest common divisor (GCD), also known as the greatest common factor (GCF) or highest common factor (HCF). The algorithm is also called Euclid's algorithm. User fills in two numbers to find out the GCD. This is a simple Numeric example, that uses While Loop inside.
Fifth-order Runge–Kutta method with adaptive step
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.
Function of the matrix (Sylvester's formula)
Computing the user-defined function of the matrix using Sylvester's formula. Example also shows how to get the coefficients of matrix characteristic polynomial with Leverrie-Faddeev method.
Computation of gravitation acceleration on the object's surface
Example demonstrates a computation of gravitation acceleration on the Solar System astronomical object's surfaces. Computation performed for eight Solar System planets and for the Sun.
Solve of tridiagonal system of equations
The tridiagonal matrix algorithm (TDMA), also known as the Thomas algorithm, is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. Example shows how to extract diagonals of the matrix and how to use it to calculate the result.
Nonlinear equations solving with dichotomy method
Nonlinear equations solving with dichotomy method. User defines initial equation to proceed, calculation precision and the range. Program returns root of the initial equation, result accuracy and number of iterations.
Hesse matrix and Hessian
Algorithm of Hesse matrix generation and the definition of the Hessian. The user specifies a function to construct Hessian matrix in the loop using partial derivatives. The last step defines the functions to work with the result. All calculations are performs in symbols, with the possibility to get Symbolic and Numeric results of the algorithm.
Laguerre polynomials solving
Solving of Laguerre polynomials. User specifies a power of the polinomial to get it's roots. Additionally represented graphs of first five Laguerre functions.