General

Elements™ Engineering-Scientific Workspace is a problem-formulating and solving tool that is very useful to mathematics, engineering, statistics, and computer science professionals. In operation, it is a matrix programming language and powerful numerical methods software package.

Elements history began in 1982.

Elements' Characteristics and Capabilities

• Unifying principles and primitives.
• Problem formulation is simple and the solution method reveals the underlying theory.
• Elements' syntax is much like mathematical literature and can be learned in minutes.
• You can declare mathematical objects explicitly by type. For example: NewVector, NewMatrix, and NewPolynomial.
• The default operation is to display object values.

          NewMatrix M[15][15] random;
          Inverse(M);
  

• All data are structured as matrices and all operations are structured as functions. An Elements program requires fewer statements than its equivalent in FORTRAN, or C. It reduces the effort of programming.
• Expressions can have any length and any number of nested or recursive function evaluations. The common operators +, -, *, /, ^ are available---also the operators vector inner product @ and cross-product #. Vectors and matrices can be subscripted to access individual elements.
• Operations can be applied to sets, scalars, vectors, quaternions, matrices, polynomials, discrete functions (table of values). When defined, operations between these data types are possible. Elements is programmed to "know" mathematical relationships.

          p = CharacteristicPolynomial(M);
          q = Fit(TableOfData);
  

• Similar operations on hypermatrices, whose elements are sets, variables, vectors, matrices, or polynomials.
• Operations on finite sets including subset, complement, Union(), and Intersection().
• Hundreds of built-in functions including all elementary functions plus Mean(), Median(), Sort(), Eigenvalues(), Integral(), PrimeDecomposition(), Gamma(), BesselJ(), and others.
• Numeric differentiation and integration of functions, including discrete functions. Optional algebraic differentiation and integration of polynomials.
• Rational fraction "m/n", real, and complex valued objects. These are specified by you or automatically created from the context of the expression being evaluated.
• Complex valued exponents.

          3/7 + 12/19;
          Sqrt(-1) ^ Sqrt(-1);
  

• Automatic computer storage allocation, which frees you from this complication.
• Program branching with if and else statements.
• Program looping with statements like for (i=0, i<10, i++).
• User-defined internal subroutines.
• Highly efficient numerical methods. Long-precision (approximately 16 decimal digits) floating-point computations. Arbitrarily high precision numbers using polynomial representation.
• Optional estimate of floating-point operations count required to solve a problem without actually performing a large number of these expensive operations.
• Built-in functions that can measure computer performance, CPU time.
• Extensive error checking including checks on syntax, non-conformability, undefined function argument, subscript range, zero divide, exponent overflow, etc. Input data verification to insure conformance with the underlying mathematical model. Error and warning messages are descriptive and point directly to the offending portion of the source code.
• Built-in debugging aids such as tracing program flow and statement execution count. A undo statement reverses the last assignment statement.
• Conversion factors between units of measure by the convert() built-in function.

          Convert(btu,calorie);
  

• Number theory functions such as PrimeDecomposition(), PrimeGT(), Gcd(), Lcm(), Tau(), Sigma(), Totient(), and ContinuedFraction().

          PrimeDecomposition(1456379371);
          ContinuedFraction(3/17);
  

• Financial and decision making functions such as PresentValue() and AmountOfAnAnnuity().
• Picard's solution of differential equations by iterative polynomial operations.
• A Gauss-Seidel solver for Laplace and Poisson partial differential equations.
• Built-in functions that compute thermodynamic variables of gaseous substances from an equation of state.
• Optimization (simplex method).
• Unlike statistical packages that provide only printed results, a descriptive statistics function value can be postprocessed.

          Proc Statistics(ObsMat);
  

• Frequency tables.
• Analysis of variance.
• Plotting of discrete and continuous functions. Surface values of a third dimension variable. Optional polar coordinates. Elements has a gallery of plot types.

• A macro facility includes source statements from your macro library.

          #include "demo1.els";
  

• Elements offers many of the advantages of APL without the special characters.
• Accompanying self-help information and demo.