SMath

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.

Solving of Hermite polynomials. User specifies a power of the polinomial to get it's roots. Additionally represented graphs of first five Hermite functions.

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.

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.

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.

Solving of Laguerre polynomials. User specifies a power of the polinomial to get it's roots. Additionally represented graphs of first five Laguerre functions.

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.

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.

Animation in SMath Studio shown by the example of planetary gear with internal teeth.

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.

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.

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.

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.

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.

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.

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.

Expansion of the user-defined function to Maclaurin series with a custom degree.

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.