Documentation

Symbolic Math Toolbox Functions

Symbolic Computations in MATLAB

Symbolic Variables, Expressions, Functions, and Preferences

argnames Input variables of symbolic function
children Subexpressions or terms of symbolic expression
disp Display symbolic input
display Display symbolic input
formula Mathematical expression defining symbolic function
funtool Function calculator
pretty Prettyprint symbolic expressions
sympref Set symbolic preferences
sym Create symbolic variables, expressions, functions, matrices
symfun Create symbolic functions
syms Shortcut for creating symbolic variables and functions
symvar Find symbolic variables in symbolic expression, matrix, or function

Operators and Elementary Operations

Arithmetic Operations

minus Symbolic subtraction
plus Symbolic addition
times Symbolic array multiplication
ldivide Symbolic array left division
rdivide Symbolic array right division
power Symbolic array power
mtimes Symbolic matrix multiplication
mldivide Symbolic matrix left division
mrdivide Symbolic matrix right division
mpower Symbolic matrix power
transpose Symbolic matrix transpose
ctranspose Symbolic matrix complex conjugate transpose
ceil Round symbolic matrix toward positive infinity
fix Round toward zero
floor Round symbolic matrix toward negative infinity
frac Symbolic matrix element-wise fractional parts
mod Symbolic modulus after division
quorem Quotient and remainder
round Symbolic matrix element-wise round
conj Symbolic complex conjugate
imag Imaginary part of complex number
real Real part of complex number
cumprod Symbolic cumulative product
cumsum Symbolic cumulative sum

Relational Operations

eq Define equation
ge Define greater than or equal to relation
gt Define greater than relation
le Define less than or equal to relation
lt Define less than relation
ne Define inequality
in Numeric type of symbolic input
isequaln Test symbolic objects for equality, treating NaN values as equal

Logical Operations

and Logical AND for symbolic expressions
not Logical NOT for symbolic expressions
or Logical OR for symbolic expressions
xor Logical XOR for symbolic expressions
all Test whether all equations and inequalities represented as elements of symbolic array are valid
any Test whether at least one of equations and inequalities represented as elements of symbolic array is valid
in Numeric type of symbolic input
isequaln Test symbolic objects for equality, treating NaN values as equal
isfinite Check whether symbolic array elements are finite
isinf Check whether symbolic array elements are infinite
isnan Check whether symbolic array elements are NaNs
logical Check validity of equation or inequality
nnz Number of nonzero elements
nonzeros Nonzero elements

Conversion Between Symbolic and Numeric

char Convert symbolic objects to strings
double Convert symbolic matrix to MATLAB numeric form
int8, int16, int32, int64 Convert symbolic matrix to signed integers
poly2sym Create symbolic polynomial from vector of coefficients
single Convert symbolic matrix to single precision
sym Create symbolic variables, expressions, functions, matrices
sym2poly Extract vector of all numeric coefficients, including zeros, from symbolic polynomial
symfun Create symbolic functions
uint8, uint16, uint32, uint64 Convert symbolic matrix to unsigned integers
vpa Variable-precision arithmetic

Mathematics

Equation Solving

Linear and Nonlinear Equations and Systems

equationsToMatrix Convert set of linear equations to matrix form
finverse Functional inverse
linsolve Solve linear system of equations given in matrix form
poles Poles of expression or function
solve Equations and systems solver
vpasolve Numeric solver

Ordinary Differential Equations and Systems

dsolve Ordinary differential equation and system solver
massMatrixForm Extract mass matrix and right side of semilinear system of differential algebraic equations
odeFunction Convert system of symbolic algebraic expressions to MATLAB function handle suitable for ode45, ode15s, and other ODE solvers
odeToVectorField Convert higher-order differential equations to systems of first-order differential equations

Differential Algebraic Equations

daeFunction Convert system of differential algebraic equations to MATLAB function handle suitable for ode15i
decic Find consistent initial conditions for first-order implicit ODE system with algebraic constraints
findDecoupledBlocks Search for decoupled blocks in systems of equations
incidenceMatrix Find incidence matrix of system of equations
isLowIndexDAE Check if differential index of system of equations is lower than 2
massMatrixForm Extract mass matrix and right side of semilinear system of differential algebraic equations
odeFunction Convert system of symbolic algebraic expressions to MATLAB function handle suitable for ode45, ode15s, and other ODE solvers
reduceDAEIndex Convert system of first-order differential algebraic equations to equivalent system of differential index 1
reduceDAEToODE Convert system of first-order semilinear differential algebraic equations to equivalent system of differential index 0
reduceDifferentialOrder Reduce system of higher-order differential equations to equivalent system of first-order differential equations
reduceRedundancies Simplify system of first-order differential algebraic equations by eliminating redundant equations and variables

Formula Manipulation and Simplification

Simplification

simplify Algebraic simplification
simplifyFraction Symbolic simplification of fractions
subexpr Rewrite symbolic expression in terms of common subexpressions

Formula Rearrangement and Rewriting

coeffs Coefficients of polynomial
collect Collect coefficients
combine Combine terms of identical algebraic structure
compose Functional composition
divisors Divisors of integer or expression
expand Symbolic expansion of polynomials and elementary functions
factor Factorization
horner Horner nested polynomial representation
numden Extract numerator and denominator
partfrac Partial fraction decomposition
rewrite Rewrite expression in new terms
subexpr Rewrite symbolic expression in terms of common subexpressions

Substitution

subexpr Rewrite symbolic expression in terms of common subexpressions
subs Symbolic substitution

Calculus

Differentiation

diff Differentiate symbolic expression or function
functionalDerivative Functional derivative

Integration

int Definite and indefinite integrals
rsums Interactive evaluation of Riemann sums

Vector Analysis

curl Curl of vector field
divergence Divergence of vector field
gradient Gradient vector of scalar function
hessian Hessian matrix of scalar function
jacobian Jacobian matrix
laplacian Laplacian of scalar function
potential Potential of vector field
vectorPotential Vector potential of vector field

Series

cumprod Symbolic cumulative product
cumsum Symbolic cumulative sum
pade Padé approximant
rsums Interactive evaluation of Riemann sums
symprod Product of series
symsum Sum of series
taylor Taylor series expansion
taylortool Taylor series calculator

Limits

limit Compute limit of symbolic expression

Transforms

fourier Fourier transform
ifourier Inverse Fourier transform
ilaplace Inverse Laplace transform
iztrans Inverse Z-transform
laplace Laplace transform
sympref Set symbolic preferences
ztrans Z-transform

Linear Algebra

Matrix Operations and Transformations

bernstein Bernstein polynomials
bernsteinMatrix Bernstein matrix
cat Concatenate symbolic arrays along specified dimension
diag Create or extract diagonals of symbolic matrices
expm Matrix exponential
horzcat Concatenate symbolic arrays horizontally
inv Compute symbolic matrix inverse
pinv Moore-Penrose inverse (pseudoinverse) of symbolic matrix
reshape Reshape symbolic array
size Symbolic matrix dimensions
sort Sort elements of symbolic vectors or matrices
toeplitz Symbolic Toeplitz matrix
tril Return lower triangular part of symbolic matrix
triu Return upper triangular part of symbolic matrix
vertcat Concatenate symbolic arrays vertically

Linear Equations

cond Condition number of matrix
det Compute determinant of symbolic matrix
equationsToMatrix Convert set of linear equations to matrix form
inv Compute symbolic matrix inverse
linsolve Solve linear system of equations given in matrix form
norm Norm of matrix or vector
rank Find rank of symbolic matrix
rref Reduced row echelon form of matrix (Gauss-Jordan elimination)

Matrix Decomposition

chol Cholesky factorization
jordan Jordan form of matrix
lu LU factorization
qr QR factorization
svd Singular value decomposition of symbolic matrix

Eigenvalues and Eigenvectors

charpoly Characteristic polynomial of matrix
eig Eigenvalues and eigenvectors of symbolic matrix
jordan Jordan form of matrix

Vector Spaces and Subspaces

colon Create symbolic vectors, array subscripting, and for-loop iterators
colspace Column space of matrix
null Form basis for null space of matrix
numel Number of elements of symbolic array
orth Orthonormal basis for range of symbolic matrix
rank Find rank of symbolic matrix
rref Reduced row echelon form of matrix (Gauss-Jordan elimination)
size Symbolic matrix dimensions

Matrix Analysis

adjoint Adjoint of symbolic square matrix
charpoly Characteristic polynomial of matrix
cond Condition number of matrix
curl Curl of vector field
det Compute determinant of symbolic matrix
divergence Divergence of vector field
gradient Gradient vector of scalar function
hessian Hessian matrix of scalar function
jacobian Jacobian matrix
laplacian Laplacian of scalar function
minpoly Minimal polynomial of matrix
norm Norm of matrix or vector
potential Potential of vector field
vectorPotential Vector potential of vector field

Matrix Functions

expm Matrix exponential
funm General matrix function
logm Matrix logarithm
sqrtm Matrix square root

Assumptions

assume Set assumption on symbolic object
assumeAlso Add assumption on symbolic object
assumptions Show assumptions affecting symbolic variable, expression, or function
clear all Remove items from MATLAB workspace and reset MuPAD engine
in Numeric type of symbolic input
isAlways Check whether equation or inequality holds for all values of its variables
isfinite Check whether symbolic array elements are finite
isinf Check whether symbolic array elements are infinite
isnan Check whether symbolic array elements are NaNs
logical Check validity of equation or inequality
reset Close MuPAD engine

Polynomials

charpoly Characteristic polynomial of matrix
chebyshevT Chebyshev polynomials of the first kind
chebyshevU Chebyshev polynomials of the second kind
coeffs Coefficients of polynomial
gegenbauerC Gegenbauer polynomials
hermiteH Hermite polynomials
jacobiP Jacobi polynomials
laguerreL Generalized Laguerre Function and Laguerre Polynomials
legendreP Legendre polynomials
minpoly Minimal polynomial of matrix
poly2sym Create symbolic polynomial from vector of coefficients
sym2poly Extract vector of all numeric coefficients, including zeros, from symbolic polynomial

Mathematical Functions

Constants

catalan Catalan constant
eulergamma Euler-Mascheroni constant

Logarithms

log Natural logarithm of entries of symbolic matrix
log10 Logarithm base 10 of entries of symbolic matrix
log2 Logarithm base 2 of entries of symbolic matrix

Trigonometric Functions

sin Symbolic sine function
cos Symbolic cosine function
tan Symbolic tangent function
cot Symbolic cotangent function
sec Symbolic secant function
csc Symbolic cosecant function
asin Symbolic inverse sine function
acos Symbolic inverse cosine function
atan Symbolic inverse tangent function
acot Symbolic inverse cotangent function
asec Symbolic inverse secant function
acsc Symbolic inverse cosecant function

Hyperbolic Functions

sinh Symbolic hyperbolic sine function
cosh Symbolic hyperbolic cosine function
tanh Symbolic hyperbolic tangent function
coth Symbolic hyperbolic cotangent function
sech Symbolic hyperbolic secant function
csch Symbolic hyperbolic cosecant function
asinh Symbolic inverse hyperbolic sine function
acosh Symbolic inverse hyperbolic cosine function
atanh Symbolic inverse hyperbolic tangent function
acoth Symbolic inverse hyperbolic cotangent function
asech Symbolic inverse hyperbolic secant function
acsch Symbolic inverse hyperbolic cosecant function

Complex Numbers

abs Absolute value of real or complex value
angle Symbolic polar angle
atan2 Symbolic four-quadrant inverse tangent
imag Imaginary part of complex number
real Real part of complex number
sign Sign of real or complex value
signIm Sign of the imaginary part of complex number

Special Functions

Dirac and Heaviside Functions
dirac Dirac delta function
heaviside Heaviside step function
kroneckerDelta Kronecker delta function
rectangularPulse Rectangular pulse function
sympref Set symbolic preferences
triangularPulse Triangular pulse function
Gamma Functions
beta Beta function
factorial Factorial function
gamma Gamma function
gammaln Logarithmic gamma function
igamma Incomplete gamma function
nchoosek Binomial coefficient
pochhammer Pochhammer symbol
Zeta Function and Polylogarithms
catalan Catalan constant
dilog Dilogarithm function
polylog Polylogarithm
psi Digamma function
zeta Riemann zeta function
Airy and Bessel Functions
airy Airy function
besseli Modified Bessel function of the first kind
besselj Bessel function of the first kind
besselk Modified Bessel function of the second kind
bessely Bessel function of the second kind
Exponential and Trigonometric Integrals
coshint Hyperbolic cosine integral function
cosint Cosine integral function
dawson Dawson integral
ei One-argument exponential integral function
eulergamma Euler-Mascheroni constant
expint Exponential integral function
fresnelc Fresnel cosine integral function
fresnels Fresnel sine integral function
logint Logarithmic integral function
sinhint Hyperbolic sine integral function
sinint Sine integral function
ssinint Shifted sine integral function
Error Functions
erf Error function
erfc Complementary error function
erfcinv Inverse complementary error function
erfi Imaginary error function
erfinv Inverse error function
Hypergeometric and Whittaker Functions
hypergeom Hypergeometric function
kummerU Confluent hypergeometric Kummer U function
whittakerM Whittaker M function
whittakerW Whittaker W function
Elliptic Integrals
ellipke Complete elliptic integrals of the first and second kinds
ellipticCE Complementary complete elliptic integral of the second kind
ellipticCK Complementary complete elliptic integral of the first kind
ellipticCPi Complementary complete elliptic integral of the third kind
ellipticE Complete and incomplete elliptic integrals of the second kind
ellipticF Incomplete elliptic integral of the first kind
ellipticK Complete elliptic integral of the first kind
ellipticPi Complete and incomplete elliptic integrals of the third kind
Lambert W and Wright Functions
lambertw Lambert W function
wrightOmega Wright omega function

Numbers and Precision

Precision Control

digits Variable-precision accuracy
double Convert symbolic matrix to MATLAB numeric form
vpa Variable-precision arithmetic

Operations on Numbers

ceil Round symbolic matrix toward positive infinity
conj Symbolic complex conjugate
fix Round toward zero
floor Round symbolic matrix toward negative infinity
frac Symbolic matrix element-wise fractional parts
imag Imaginary part of complex number
max Largest elements
min Smallest elements
real Real part of complex number
round Symbolic matrix element-wise round

Number Theory

bernoulli Bernoulli numbers and polynomials
divisors Divisors of integer or expression
euler Euler numbers and polynomials
gcd Greatest common divisor
harmonic Harmonic function (harmonic number)
lcm Least common multiple
mod Symbolic modulus after division
quorem Quotient and remainder
rem Remainder after division

Graphics

Function Plots

ezcontour Contour plotter
ezcontourf Filled contour plotter
ezmesh 3-D mesh plotter
ezmeshc Combined mesh and contour plotter
ezplot Plot symbolic expression, equation, or function
ezplot3 3-D parametric curve plotter
ezpolar Polar coordinate plotter
ezsurf Plot 3-D surface
ezsurfc Combined surface and contour plotter

Code Generation

ccode C code representation of symbolic expression
fortran Fortran representation of symbolic expression
latex LaTeX representation of symbolic expression
matlabFunction Convert symbolic expression to function handle or file
matlabFunctionBlock Convert symbolic expression to MATLAB Function block
simscapeEquation Convert symbolic expressions to Simscape language equations
texlabel TeX representation of symbolic expression

MATLAB and MuPAD Integration

MuPAD Function Calls

evalin Evaluate MuPAD expressions without specifying their arguments
feval Evaluate MuPAD expressions specifying their arguments
read Read MuPAD program file into symbolic engine

MuPAD Files and Interfaces

allMuPADNotebooks All open notebooks
close Close MuPAD notebook
doc Get help for MuPAD functions
evaluateMuPADNotebook Evaluate MuPAD notebook
mupad Start MuPAD notebook
mupadNotebookTitle Window title of MuPAD notebook
mupadwelcome Start MuPAD interfaces
openmn Open MuPAD notebook
openmu Open MuPAD program file
openxvc Open MuPAD uncompressed graphics file (XVC)
openxvz Open MuPAD compressed graphics file (XVZ)

Variables and Expressions Exchange

getVar Get variable from MuPAD notebook
setVar Assign variable in MuPAD notebook

MuPAD Engine Commands

clear all Remove items from MATLAB workspace and reset MuPAD engine
reset Close MuPAD engine
symengine Return symbolic engine
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