Symbolic Vectors, Matrices and Arrays
By using a symbol to represent a vector, matrix or array, one gets an efficient notation to model a mathematical problem. Indeed, most scientific, engineering and statistical domains have transitioned to use this type of more abstract and efficient notation. The Wolfram Language has a rich symbolic array language to describe problems. Most high-level solvers support symbolic array expressions and array variables, making it easy and efficient to specify high-dimensional problems.
Symbolic Array Variables
x∈Vectors[…] — assume x is a vector
x∈region — assume x is a vector from a geometric region
VectorSymbol — define a vector symbol that can be used together with listable functions
MatrixSymbol ▪ ArraySymbol ▪ NonThreadable
Symbolic Array Constants
Common zero-one arrays in array formulas.
SymbolicZerosArray ▪ SymbolicOnesArray ▪ SymbolicIdentityArray ▪ SymbolicDeltaProductArray
Symbolic Array Functions
Dot — vector and matrix inner product
ArrayDot — generalized array inner product
Norm ▪ Tr ▪ Det ▪ Cross ▪ Transpose ▪ TensorProduct ▪ TensorContract ▪ KroneckerProduct ▪ TensorWedge
Inverse ▪ Adjugate ▪ PseudoInverse ▪ LinearSolve ▪ LeastSquares ▪ MatrixPower ▪ MatrixExp ▪ MatrixLog ▪ MatrixFunction
Total ▪ Mean ▪ StandardDeviation ▪ Variance ▪ Covariance ▪ Correlation ▪ AbsoluteCorrelation ▪ Kurtosis ▪ Skewness ▪ Moment ▪ CentralMoment ▪ FactorialMoment ▪ Cumulant
Symbolic Array Predicates
Array equations and inequations
VectorLessEqual ▪ VectorLess ▪ VectorGreaterEqual ▪ VectorGreater
Simplification and Transformations
ArraySimplify — simplify symbolic array expressions
ArrayExpand — expand symbolic array expressions
ComponentExpand — expand symbolic array expressions into their components
Array Derivatives
D — symbolic differentiation w.r.t. vector, matrix and array variables
Array Limits
Limit — compute limits using symbolic vector variables
Array Algebraic Equation Solvers
Solve — solve equations and inequalities with symbolic array variables
NSolve ▪ SolveValues ▪ NSolveValues ▪ Reduce ▪ FindInstance ▪ FindRoot
Array Optimization Solvers
Minimize — optimize using symbolic array variables
MinValue ▪ ArgMin ▪ Maximize ▪ MaxValue ▪ ArgMax ▪ NMinimize ▪ NMinValue ▪ NArgMin ▪ NMaximize ▪ NMaxValue ▪ NArgMax ▪ FindMinimum ▪ FindMinValue ▪ FindArgMin ▪ FindMaximum ▪ FindMaxValue ▪ FindArgMax
Convex optimization constraints are often expressed using vector and matrix inequalities.
ConvexOptimization ▪ ParametricConvexOptimization ▪ RobustConvexOptimization ▪ LinearOptimization ▪ LinearFractionalOptimization ▪ QuadraticOptimization ▪ SecondOrderConeOptimization ▪ SemidefiniteOptimization ▪ GeometricOptimization ▪ ConicOptimization
Array Integration Solvers
Integrate, NIntegrate — integrate expressions with symbolic vector variables
LineIntegrate ▪ NLineIntegrate ▪ SurfaceIntegrate ▪ NSurfaceIntegrate
Array Differential Equation Solvers
NDSolve — solve differential equations with symbolic array variables
DSolve ▪ NDSolveValue ▪ DSolveValue ▪ ParametricNDSolveValue ▪ AsymptoticDSolveValue
Array Difference Equation Solvers
RSolve — solve difference equations with symbolic array variables