MATLAB Symbolic Math Toolbox Toolbox™ User's Guide

Description

Getting Started Symbolic Math Toolbox Product Description Key Features Create Symbolic Numbers, Variables, and Expressions Create Symbolic Numbers Create Symbolic Variables Create Symbolic Expressions Reuse Names of Symbolic Objects Create Symbolic Functions Create Symbolic Matrices Use Existing Symbolic Variables Generate Elements While Creating a Matrix Create Matrix of Symbolic Numbers Perform Symbolic Computations Differentiate Symbolic Expressions Integrate Symbolic Expressions Solve Equations Simplify Symbolic Expressions Substitutions in Symbolic Expressions Plot Symbolic Functions Use Assumptions on Symbolic Variables Default Assumption Set Assumptions Check Existing Assumptions Delete Symbolic Objects and Their Assumptions Using Symbolic Math Toolbox Software Find Symbolic Variables in Expressions, Functions, Matrices Find a Default Symbolic Variable Choose syms or sym Function Add Subscripts, Superscripts, and Accents to Symbolic Variables Change Output Display Format of Symbolic Results Copy and Paste Symbolic Output in Live Editor Units of Measurement Tutorial Define and Convert Units Use Temperature Units in Absolute or Difference Forms Verify Dimensions Use Units in Differential Equations Units and Unit Systems List Units List SI Unit Prefixes List Unit Systems List Defining Constants of SI Units Unit Conversions and Unit Systems Convert Units Temperature Unit Conversion Convert to SI, CGS, or US Unit Systems Define Custom Unit System from Existing System Define Custom Unit System Directly Unit System Definition Differentiation Derivatives of Expressions with Several Variables More Examples Functional Derivatives Tutorial Limits One-Sided Limits Integration Integration with Real Parameters Integration with Complex Parameters High-Precision Numerical Integration Using Variable-Precision Arithmetic Symbolic Summation Comparing symsum and sum Computational Speed of symsum versus sum Output Format Differences Between symsum and sum Taylor Series Padé Approximant Find Asymptotes, Critical, and Inflection Points Simplify Symbolic Expressions Simplify Using Options Simplify Using Assumptions Simplify Fractions Solve Algebraic Equation Using Live Editor Task Simplify Symbolic Expressions Using Live Editor Task Abbreviate Common Terms in Long Expressions Choose Function to Rearrange Expression Combine Terms of Same Algebraic Structures Expand Expressions Factor Expressions Extract Subexpressions from Expression Collect Terms with Same Powers Rewrite Expressions in Terms of Other Functions Compute Partial Fraction Decompositions of Expressions Compute Normal Forms of Rational Expressions Represent Polynomials Using Horner Nested Forms Clear Assumptions and Reset the Symbolic Engine Check Assumptions Set on Variables Effects of Assumptions on Computations Extract Numerators and Denominators of Rational Expressions Substitute Variables in Symbolic Expressions Substitute Elements in Symbolic Matrices Substitute Scalars with Matrices Evaluate Symbolic Expressions Using subs Choose Numeric or Symbolic Arithmetic Double-Precision Arithmetic Variable-Precision Arithmetic Symbolic Arithmetic Comparisons of Numeric and Symbolic Arithmetic Increase Precision of Numeric Calculations Recognize and Avoid Round-Off Errors Use Symbolic Computations When Possible Perform Calculations with Increased Precision Compare Symbolic and Numeric Results Plot the Function or Expression Increase Speed by Reducing Precision Numeric to Symbolic Conversion Conversion to Rational Symbolic Form Conversion by Using Floating-Point Expansion Conversion to Rational Symbolic Form with Error Term Conversion to Decimal Form Basic Algebraic Operations Linear Algebraic Operations Symbolic Hilbert Matrix Symbolic Linear Algebra Operations Variable-Precision Arithmetic Symbolic Investigation of Singular Value Eigenvalues Jordan Canonical Form Singular Value Decomposition Solve Algebraic Equation Solve an Equation Return the Full Solution to an Equation Work with the Full Solution, Parameters, and Conditions Returned by solve Visualize and Plot Solutions Returned by solve Simplify Complicated Results and Improve Performance Solve a Second-Order Differential Equation Numerically Select Numeric or Symbolic Solver Solve System of Algebraic Equations Handle the Output of solve Solve a Linear System of Equations Return the Full Solution of a System of Equations Solve a System of Equations Under Conditions Work with Solutions, Parameters, and Conditions Returned by solve Convert Symbolic Results to Numeric Values Simplify Complicated Results and Improve Performance Troubleshoot Equation Solutions from solve Function Return Only Real Solutions Apply Simplification Rules Use Assumptions to Narrow Results Simplify Solutions Tips Solve System of Linear Equations Solve System of Linear Equations Using linsolve Solve System of Linear Equations Using solve Solve Equations Numerically Solve Differential Equation First-Order Linear ODE Solve Differential Equation with Condition Nonlinear Differential Equation with Initial Condition Second-Order ODE with Initial Conditions Third-Order ODE with Initial Conditions More ODE Examples Solve a System of Differential Equations Solve System of Differential Equations Solve Differential Equations in Matrix Form Solve Differential Algebraic Equations (DAEs) Solve Semilinear DAE System Step 1. Reduce Differential Index with reduceDAEToODE Step 2. ODEs to Function Handles for ode15s and ode23t Step 3. Initial Conditions for ode15s and ode23t Step 4. Solve an ODE System with ode15s or ode23t Solve DAEs Using Mass Matrix Solvers Step 1. Convert DAEs to Function Handles Step 2. Find Initial Conditions Step 3. Solve DAE System Fourier and Inverse Fourier Transforms Fourier Transform Definition Concept: Using Symbolic Workflows Calculate Beam Deflection Using Fourier Transform Solve Differential Equations Using Laplace Transform Definition: Laplace Transform Concept: Using Symbolic Workflows Workflow: Solve RLC Circuit Using Laplace Transform Solve Difference Equations Using Z-Transform Definition: Z-transform Concept: Using Symbolic Workflows Workflow: Solve "Rabbit Growth" Problem Using Z-Transform References Create Plots Plot with Symbolic Plotting Functions Plot Functions Numerically Plot Multiple Symbolic Functions in One Graph Plot Multiple Symbolic Functions in One Figure Combine Symbolic Function Plots and Numeric Data Plots Combine Numeric and Symbolic Plots in 3-D Plotting in Spherical Coordinate System Analytical Solutions of the Inverse Kinematics of a Humanoid Robot Analytical Model of Cantilever Truss Structure for Simscape Estimate Model Parameters of a Symbolically Derived Plant Model in Simulink Generate C or Fortran Code from Symbolic Expressions Generate MATLAB Functions from Symbolic Expressions Generating a Function Handle Control the Order of Variables Generate a File Name Output Variables Generate MATLAB Function Blocks from Symbolic Expressions Generate and Edit a Block Control the Order of Input Ports Name the Output Ports Generate Simscape Equations from Symbolic Expressions Convert Algebraic and Differential Equations Limitations MuPAD to MATLAB Migration MuPAD Engines and MATLAB Workspace Convert MuPAD Notebooks to MATLAB Live Scripts Convert a MuPAD Notebook .mn to a MATLAB Live Script .mlx Convert MuPAD Graphics to MATLAB Graphics Known Issues Troubleshoot MuPAD to MATLAB Translation Errors Troubleshoot MuPAD to MATLAB Translation Warnings Differences Between MATLAB and MuPAD Syntax Call Built-In MuPAD Functions from MATLAB evalin feval evalin vs. feval Floating-Point Arguments of evalin and feval Functions abs acos acosh acot acoth acsc acsch adjoint airy all and angle animationToFrame Animator any argnames asec asech asin asinh assume assumeAlso assumptions atan atan2 atanh baseUnits bernoulli bernstein bernsteinMatrix besselh besseli besselj besselk bessely beta cat catalan ccode cell2sym changeIntegrationVariable charpoly chebyshevT chebyshevU checkUnits children chol coeffs collect colon colspace combine compose cond conj convertMuPADNotebook cos cosh coshint cosint cot coth csc csch ctranspose cumprod cumsum curl daeFunction dawson dec2bin dec2hex decic derivedUnits det diag diff digits dilog dirac displayFormula divergence divisors double dsolve ei eig eliminate ellipke ellipticCE ellipticCK ellipticCPi ellipticE ellipticF ellipticK ellipticNome ellipticPi eq equationsToMatrix erf erfc erfcinv erfi erfinv euler eulergamma eulerPhi evalin expand expint expm ezcontour ezcontourf ezmesh ezmeshc ezplot ezplot3 ezpolar ezsurf ezsurfc factor factorial factorIntegerPower fanimator fcontour feval fibonacci fimplicit fimplicit3 findDecoupledBlocks findSymType findUnits finverse fmesh fold formula fortran fourier fplot fplot3 fresnelc fresnels fsurf functionalDerivative funm funtool gamma gammaln gcd ge gegenbauerC gradient gbasis gt harmonic has hasSymType heaviside hermiteForm hermiteH hessian horner horzcat htrans hurwitzZeta hypergeom ifourier igamma ihtrans ilaplace imag in incidenceMatrix int integrateByParts inv isAlways isequal isequaln isfinite isinf isLowIndexDAE isnan isolate isPrimitiveRoot isSymType isUnit iztrans jacobiAM jacobian jacobiCD jacobiCN jacobiCS jacobiDC jacobiDN jacobiDS jacobiNC jacobiND jacobiNS jacobiP jacobiSC jacobiSD jacobiSN jacobiSymbol jacobiZeta jordan kroneckerDelta kummerU laguerreL lambertw laplace laplacian latex lcm ldivide le legendreP lhs limit linsolve log log10 log2 logical logint logm lt lu mapSymType massMatrixForm mathml matlabFunction matlabFunctionBlock meijerG mfun mfunlist minpoly minus mixedUnits mldivide mod mpower mrdivide mtimes mupad nchoosek ne newUnit newUnitSystem nextprime norm not nthprime nthroot null numden odeFunction odeToVectorField or orth pade partfrac piecewise pinv playAnimation plus pochhammer poles poly2sym polylog polynomialDegree polynomialReduce potential power powermod pretty prevprime psi qr quorem rank rat rdivide read real rectangularPulse reduceDAEIndex reduceDAEToODE reduceDifferentialOrder reduceRedundancies release rem removeUnit reset reshape resultant removeUnitSystem rewindAnimation rewrite rhs root rref rsums sec sech separateUnits series sign signIm simplify simplifyFraction Simplify Symbolic Expression simscapeEquation sin sinc sinh sinhint sinint smithForm solve Solve Symbolic Equation sort sqrtm ssinint str2symunit str2sym subexpr subs svd sym sym2cell sym2poly symengine symfalse symfun symFunType sympref symprod symReadSSCParameters symReadSSCVariables syms symsum symtrue symType symunit symunit2str symvar symWriteSSC tan tanh taylor taylortool texlabel times toeplitz transpose triangularPulse tril triu unitConversionFactor unitConvert unitInfo unitSystems vectorPotential vertcat vpa vpaintegral vpasolve whittakerM whittakerW wrightOmega writeAnimation xor zeta ztrans