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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.

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.

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.

Generation of correlated random variable sets with normal distribution

Generates correlated random variables from uncorrelated random variables with normal distribution

Beam FEA with interactive preprocessor

2D horizontal beam finite element analysis with exact representation of trapezoidal distributed loads. Features an interactive preprocessor and graphic and tabular output

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.

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.

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.

Expansion of function to Maclaurin series

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

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.

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.

Text region Fonts

A collection of text regions with different font-families.

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 Uniits.

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.

Colors of the rainbow

Game to compose colors of the rainbow in correct order.

Properties of generic polygons

calculate properties of generic polygons:
- perimeter
- area
- centroid
- second moment of area
- radii of gyration
- elastic section modulus
- plastic section modulus
- orientation of principal axes of inertia
- principal moments of inertia
- radius of gyration about principal axes of inertia
- shortcuts for easy plots
- multilanguage [EN/IT]

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.

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.

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.

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.

Foundation calculations

The program of calculation of monolithic reinforced concrete foundations. Calculates the concrete foundation under static loading of an arbitrary number of columns. Creates a graphical representation with the preparation of specifications and steel sampling costs.

Embedded Plate calculations

Embedded Plate calculation program. Plate is calculated under static loading. Creates a graphical representation and prepares specifications.

2D plots with hatch and fill

This example demonstrates the usage of the snippets hatch and fill in 2D plots

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.

Planetary gear with internal teeth

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

Language-Integrated Query (Linq).

Useful functions for working with sets.

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.

Solving Linear Systems of Equations with the Cholesky method

Example for the solution of linear systems of equations with positiv definite coefficient matrix, via the Cholesky method

Steam engine

Oscillating cylinder steam engine

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.