Mathematics :: Function Reference (MATLAB®)

Arrays and Matrices	Basic array operators and operations, creation of elementary and specialized arrays and matrices
Linear Algebra	Matrix analysis, linear equations, eigenvalues, singular values, logarithms, exponentials, factorization
Elementary Math	Trigonometry, exponentials and logarithms, complex values, rounding, remainders, discrete math
Polynomials	Multiplication, division, evaluation, roots, derivatives, integration, eigenvalue problem, curve fitting, partial fraction expansion
Interpolation and Computational Geometry	Interpolation, Delaunay triangulation, convex hulls, Voronoi diagrams, domain generation
Cartesian Coordinate System Conversion	Conversions between Cartesian and polar or spherical coordinates
Nonlinear Numerical Methods	Differential equations, optimization, integration
Special Functions	Airy, Bessel, Jacobi, Legendre, beta, elliptic, error, exponential integral, gamma functions
Sparse Matrices	Elementary sparse matrices, operations, reordering algorithms, linear algebra, iterative methods, tree operations
Math Constants	Pi, imaginary unit, infinity, Not-a-Number, largest and smallest positive floating point numbers, floating point relative accuracy

Arrays and Matrices

Basic Information	Display array contents, get array information, determine array type
Operators	Arithmetic operators
Elementary Matrices and Arrays	Create elementary arrays of different types, generate arrays for plotting, array indexing, etc.
Array Operations	Operate on array content, apply function to each array element, find cumulative product or sum, etc.
Array Manipulation	Create, sort, rotate, permute, reshape, and shift array contents
Specialized Matrices	Create Hadamard, Companion, Hankel, Vandermonde, Pascal matrices, etc.

Basic Information

disp	Display text or array
display	Display text or array (overloaded method)
iscolumn	Determine whether input is column vector
isempty	Determine whether array is empty
isequal	Test arrays for equality
isequalwithequalnans	Test arrays for equality, treating NaNs as equal
isfinite	Array elements that are finite
isfloat	Determine whether input is floating-point array
isinf	Array elements that are infinite
isinteger	Determine whether input is integer array
islogical	Determine whether input is logical array
ismatrix	Determine whether input is matrix
isnan	Array elements that are NaN
isnumeric	Determine whether input is numeric array
isrow	Determine whether input is row vector
isscalar	Determine whether input is scalar
issparse	Determine whether input is sparse
isvector	Determine whether input is vector
length	Length of vector or largest array dimension
max	Largest elements in array
min	Smallest elements in array
ndims	Number of array dimensions
numel	Number of elements in array or subscripted array expression
size	Array dimensions

Operators

Arithmetic Operators + - * / \ ^ '	Matrix and array arithmetic
ctranspose	Complex conjugate transpose
minus	Minus
mpower	Matrix power
plus	Plus
times	Array multiply
transpose	Transpose
uminus	Unary minus
uplus	Unary plus

+	Addition
+	Unary plus
-	Subtraction
-	Unary minus
*	Matrix multiplication
^	Matrix power
\	Backslash or left matrix divide
/	Slash or right matrix divide
'	Transpose
.'	Nonconjugated transpose
.*	Array multiplication (element-wise)
.^	Array power (element-wise)
.\	Left array divide (element-wise)
./	Right array divide (element-wise)

Elementary Matrices and Arrays

blkdiag	Construct block diagonal matrix from input arguments
colon (:)	Create vectors, array subscripting, and `for`-loop iterators
diag	Diagonal matrices and diagonals of matrix
eye	Identity matrix
freqspace	Frequency spacing for frequency response
ind2sub	Subscripts from linear index
linspace	Generate linearly spaced vectors
logspace	Generate logarithmically spaced vectors
ndgrid	Rectangular grid in N-D space
ones	Create array of all ones
rand	Uniformly distributed pseudorandom numbers
randi	Uniformly distributed pseudorandom integers
randn	Normally distributed pseudorandom numbers
RandStream	Random number stream
rng	Control random number generation
sub2ind	Convert subscripts to linear indices
zeros	Create array of all zeros

Array Operations

See Linear Algebra and Elementary Math for other array operations.

accumarray	Construct array with accumulation
arrayfun	Apply function to each element of array
bsxfun	Apply element-by-element binary operation to two arrays with singleton expansion enabled
cast	Cast variable to different data type
cross	Vector cross product
cumprod	Cumulative product
cumsum	Cumulative sum
dot	Vector dot product
idivide	Integer division with rounding option
kron	Kronecker tensor product
prod	Product of array elements
sum	Sum of array elements
surfnorm	Compute and display 3-D surface normals
tril	Lower triangular part of matrix
triu	Upper triangular part of matrix

Array Manipulation

blkdiag	Construct block diagonal matrix from input arguments
cat	Concatenate arrays along specified dimension
circshift	Shift array circularly
diag	Diagonal matrices and diagonals of matrix
end	Terminate block of code, or indicate last array index
flipdim	Flip array along specified dimension
fliplr	Flip matrix left to right
flipud	Flip matrix up to down
horzcat	Concatenate arrays horizontally
inline	Construct inline object
ipermute	Inverse permute dimensions of N-D array
permute	Rearrange dimensions of N-D array
repmat	Replicate and tile array
reshape	Reshape array
rot90	Rotate matrix 90 degrees
shiftdim	Shift dimensions
sort	Sort array elements in ascending or descending order
sortrows	Sort rows in ascending order
squeeze	Remove singleton dimensions
vectorize	Vectorize expression
vertcat	Concatenate arrays vertically

Specialized Matrices

compan	Companion matrix
gallery	Test matrices
hadamard	Hadamard matrix
hankel	Hankel matrix
hilb	Hilbert matrix
invhilb	Inverse of Hilbert matrix
magic	Magic square
pascal	Pascal matrix
rosser	Classic symmetric eigenvalue test problem
toeplitz	Toeplitz matrix
vander	Vandermonde matrix
wilkinson	Wilkinson's eigenvalue test matrix

Linear Algebra

Matrix Analysis	Compute norm, rank, determinant, condition number, etc.
Linear Equations	Solve linear systems, least squares, LU factorization, Cholesky factorization, etc.
Eigenvalues and Singular Values	Eigenvalues, eigenvectors, Schur decomposition, Hessenburg matrices, etc.
Matrix Logarithms and Exponentials	Matrix logarithms, exponentials, square root
Factorization	Cholesky, LU, and QR factorizations, diagonal forms, singular value decomposition

Matrix Analysis

cond	Condition number with respect to inversion
condeig	Condition number with respect to eigenvalues
det	Matrix determinant
norm	Vector and matrix norms
normest	2-norm estimate
null	Null space
orth	Range space of matrix
rank	Rank of matrix
rcond	Matrix reciprocal condition number estimate
rref	Reduced row echelon form
subspace	Angle between two subspaces
trace	Sum of diagonal elements

Linear Equations

chol	Cholesky factorization
cholinc	Sparse incomplete Cholesky and Cholesky-Infinity factorizations
cond	Condition number with respect to inversion
condest	1-norm condition number estimate
funm	Evaluate general matrix function
ichol	Incomplete Cholesky factorization
ilu	Sparse incomplete LU factorization
inv	Matrix inverse
ldl	Block LDL' factorization for Hermitian indefinite matrices
linsolve	Solve linear system of equations
lscov	Least-squares solution in presence of known covariance
lsqnonneg	Solve nonnegative least-squares constraints problem
lu	LU matrix factorization
luinc	Sparse incomplete LU factorization
pinv	Moore-Penrose pseudoinverse of matrix
qr	Orthogonal-triangular decomposition
rcond	Matrix reciprocal condition number estimate

Eigenvalues and Singular Values

balance	Diagonal scaling to improve eigenvalue accuracy
cdf2rdf	Convert complex diagonal form to real block diagonal form
condeig	Condition number with respect to eigenvalues
eig	Eigenvalues and eigenvectors
eigs	Largest eigenvalues and eigenvectors of matrix
gsvd	Generalized singular value decomposition
hess	Hessenberg form of matrix
ordeig	Eigenvalues of quasitriangular matrices
ordqz	Reorder eigenvalues in QZ factorization
ordschur	Reorder eigenvalues in Schur factorization
poly	Polynomial with specified roots
polyeig	Polynomial eigenvalue problem
rsf2csf	Convert real Schur form to complex Schur form
schur	Schur decomposition
sqrtm	Matrix square root
ss2tf	Convert state-space filter parameters to transfer function form
svd	Singular value decomposition
svds	Find singular values and vectors

Matrix Logarithms and Exponentials

expm	Matrix exponential
logm	Matrix logarithm
sqrtm	Matrix square root

Factorization

balance	Diagonal scaling to improve eigenvalue accuracy
cdf2rdf	Convert complex diagonal form to real block diagonal form
chol	Cholesky factorization
cholinc	Sparse incomplete Cholesky and Cholesky-Infinity factorizations
cholupdate	Rank 1 update to Cholesky factorization
gsvd	Generalized singular value decomposition
ichol	Incomplete Cholesky factorization
ilu	Sparse incomplete LU factorization
ldl	Block LDL' factorization for Hermitian indefinite matrices
lu	LU matrix factorization
luinc	Sparse incomplete LU factorization
planerot	Givens plane rotation
qr	Orthogonal-triangular decomposition
qrdelete	Remove column or row from QR factorization
qrinsert	Insert column or row into QR factorization
qrupdate	Rank 1 update to QR factorization
qz	QZ factorization for generalized eigenvalues
rsf2csf	Convert real Schur form to complex Schur form
svd	Singular value decomposition

Elementary Math

Trigonometric	Trigonometric functions with results in radians or degrees
Exponential	Exponential, logarithm, power, and root functions
Complex	Numbers with real and imaginary components, phase angles
Rounding and Remainder	Rounding, modulus, and remainder
Discrete Math	Prime factors, factorials, permutations, rational fractions, least common multiple, greatest common divisor

Trigonometric

acos	Inverse cosine; result in radians
acosd	Inverse cosine; result in degrees
acosh	Inverse hyperbolic cosine
acot	Inverse cotangent; result in radians
acotd	Inverse cotangent; result in degrees
acoth	Inverse hyperbolic cotangent
acsc	Inverse cosecant; result in radians
acscd	Inverse cosecant; result in degrees
acsch	Inverse hyperbolic cosecant
asec	Inverse secant; result in radians
asecd	Inverse secant; result in degrees
asech	Inverse hyperbolic secant
asin	Inverse sine; result in radians
asind	Inverse sine; result in degrees
asinh	Inverse hyperbolic sine
atan	Inverse tangent; result in radians
atan2	Four-quadrant inverse tangent
atand	Inverse tangent; result in degrees
atanh	Inverse hyperbolic tangent
cos	Cosine of argument in radians
cosd	Cosine of argument in degrees
cosh	Hyperbolic cosine
cot	Cotangent of argument in radians
cotd	Cotangent of argument in degrees
coth	Hyperbolic cotangent
csc	Cosecant of argument in radians
cscd	Cosecant of argument in degrees
csch	Hyperbolic cosecant
hypot	Square root of sum of squares
sec	Secant of argument in radians
secd	Secant of argument in degrees
sech	Hyperbolic secant
sin	Sine of argument in radians
sind	Sine of argument in degrees
sinh	Hyperbolic sine of argument in radians
tan	Tangent of argument in radians
tand	Tangent of argument in degrees
tanh	Hyperbolic tangent

Exponential

exp	Exponential
expm1	Compute exp(x)-1 accurately for small values of x
log	Natural logarithm
log10	Common (base 10) logarithm
log1p	Compute log(1+x) accurately for small values of x
log2	Base 2 logarithm and dissect floating-point numbers into exponent and mantissa
nextpow2	Next higher power of 2
nthroot	Real nth root of real numbers
pow2	Base 2 power and scale floating-point numbers
power	Array power
reallog	Natural logarithm for nonnegative real arrays
realpow	Array power for real-only output
realsqrt	Square root for nonnegative real arrays
sqrt	Square root

Complex

abs	Absolute value and complex magnitude
angle	Phase angle
complex	Construct complex data from real and imaginary components
conj	Complex conjugate
cplxpair	Sort complex numbers into complex conjugate pairs
i	Imaginary unit
imag	Imaginary part of complex number
isreal	Check if input is real array
j	Imaginary unit
real	Real part of complex number
sign	Signum function
unwrap	Correct phase angles to produce smoother phase plots

Rounding and Remainder

ceil	Round toward positive infinity
fix	Round toward zero
floor	Round toward negative infinity
idivide	Integer division with rounding option
mod	Modulus after division
rem	Remainder after division
round	Round to nearest integer

Discrete Math

factor	Prime factors
factorial	Factorial function
gcd	Greatest common divisor
isprime	Array elements that are prime numbers
lcm	Least common multiple
nchoosek	Binomial coefficient or all combinations
perms	All possible permutations
primes	Generate list of prime numbers
rat, rats	Rational fraction approximation

Polynomials

conv	Convolution and polynomial multiplication
deconv	Deconvolution and polynomial division
poly	Polynomial with specified roots
polyder	Polynomial derivative
polyeig	Polynomial eigenvalue problem
polyfit	Polynomial curve fitting
polyint	Integrate polynomial analytically
polyval	Polynomial evaluation
polyvalm	Matrix polynomial evaluation
residue	Convert between partial fraction expansion and polynomial coefficients
roots	Polynomial roots

Interpolation and Computational Geometry

Interpolation	Data interpolation, data gridding, polynomial evaluation, nearest point search
Delaunay Triangulation	Delaunay triangulation, triangular surface and mesh plots
Convex Hull	Plot convex hull, plotting functions
Voronoi Diagrams	Plot Voronoi diagram, patch graphics object, plotting functions
Domain Generation	Generate arrays for 3-D plots, or for N-D functions and interpolation

Interpolation

dsearch	Search Delaunay triangulation for nearest point
dsearchn	N-D nearest point search
griddata	Data gridding
griddata3	Data gridding and hypersurface fitting for 3-D data
griddatan	Data gridding and hypersurface fitting (dimension ≥ 2)
griddedInterpolant	Interpolant for gridded data
interp1	1-D data interpolation (table lookup)
interp1q	Quick 1-D linear interpolation
interp2	2-D data interpolation (table lookup)
interp3	3-D data interpolation (table lookup)
interpft	1-D interpolation using FFT method
interpn	N-D data interpolation (table lookup)
meshgrid	Rectangular grid in 2-D and 3-D space
mkpp	Make piecewise polynomial
ndgrid	Rectangular grid in N-D space
padecoef	Padé approximation of time delays
pchip	Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)
ppval	Evaluate piecewise polynomial
spline	Cubic spline data interpolation
TriScatteredInterp	Interpolate scattered data
tsearch	Search for enclosing Delaunay triangle
tsearchn	N-D closest simplex search
unmkpp	Piecewise polynomial details

Delaunay Triangulation

delaunay	Delaunay triangulation
delaunayn	N-D Delaunay triangulation
size (TriRep)	Size of triangulation matrix
trimesh	Triangular mesh plot
triplot	2-D triangular plot
TriRep	Triangulation representation
trisurf	Triangular surface plot

Convex Hull

convexHull (DelaunayTri)	Convex hull
convhull	Convex hull
convhulln	N-D convex hull
patch	Create one or more filled polygons
trisurf	Triangular surface plot

Voronoi Diagrams

patch	Create one or more filled polygons
voronoi	Voronoi diagram
voronoiDiagram (DelaunayTri)	Voronoi diagram
voronoin	N-D Voronoi diagram

Domain Generation

meshgrid	Rectangular grid in 2-D and 3-D space
ndgrid	Rectangular grid in N-D space

Cartesian Coordinate System Conversion

cart2pol	Transform Cartesian coordinates to polar or cylindrical
cart2sph	Transform Cartesian coordinates to spherical
pol2cart	Transform polar or cylindrical coordinates to Cartesian
sph2cart	Transform spherical coordinates to Cartesian

Nonlinear Numerical Methods

Ordinary Differential Equations	Solve stiff and nonstiff differential equations, define the problem, set solver options, evaluate solution
Delay Differential Equations	Solve delay differential equations with constant and general delays, set solver options, evaluate solution
Boundary Value Problems	Solve boundary value problems for ordinary differential equations, set solver options, evaluate solution
Partial Differential Equations	Solve initial-boundary value problems for parabolic-elliptic PDEs, evaluate solution
Optimization	Find minimum of single and multivariable functions, solve nonnegative least-squares constraint problem
Numerical Integration (Quadrature)	Evaluate Simpson, Lobatto, and vectorized quadratures, evaluate double and triple integrals

Ordinary Differential Equations

decic	Compute consistent initial conditions for ode15i
deval	Evaluate solution of differential equation problem
ode15i	Solve fully implicit differential equations, variable order method
ode23, ode45, ode113, ode15s, ode23s, ode23t, ode23tb	Solve initial value problems for ordinary differential equations
odeget	Ordinary differential equation options parameters
odeset	Create or alter options structure for ordinary differential equation solvers
odextend	Extend solution of initial value problem for ordinary differential equation

Delay Differential Equations

dde23	Solve delay differential equations (DDEs) with constant delays
ddeget	Extract properties from delay differential equations options structure
ddesd	Solve delay differential equations (DDEs) with general delays
ddeset	Create or alter delay differential equations options structure
deval	Evaluate solution of differential equation problem

Boundary Value Problems

bvp4c	Solve boundary value problems for ordinary differential equations
bvp5c	Solve boundary value problems for ordinary differential equations
bvpget	Extract properties from options structure created with bvpset
bvpinit	Form initial guess for bvp4c
bvpset	Create or alter options structure of boundary value problem
bvpxtend	Form guess structure for extending boundary value solutions
deval	Evaluate solution of differential equation problem

Partial Differential Equations

pdepe	Solve initial-boundary value problems for parabolic-elliptic PDEs in 1-D
pdeval	Evaluate numerical solution of PDE using output of pdepe

Optimization

fminbnd	Find minimum of single-variable function on fixed interval
fminsearch	Find minimum of unconstrained multivariable function using derivative-free method
fzero	Find root of continuous function of one variable
lsqnonneg	Solve nonnegative least-squares constraints problem
optimget	Optimization options values
optimset	Create or edit optimization options structure

Numerical Integration (Quadrature)

dblquad	Numerically evaluate double integral over rectangle
quad	Numerically evaluate integral, adaptive Simpson quadrature
quad2d	Numerically evaluate double integral over planar region
quadgk	Numerically evaluate integral, adaptive Gauss-Kronrod quadrature
quadl	Numerically evaluate integral, adaptive Lobatto quadrature
quadv	Vectorized quadrature
triplequad	Numerically evaluate triple integral

Special Functions

airy	Airy functions
besselh	Bessel function of third kind (Hankel function)
besseli	Modified Bessel function of first kind
besselj	Bessel function of first kind
besselk	Modified Bessel function of second kind
bessely	Bessel function of second kind
beta	Beta function
betainc	Incomplete beta function
betaincinv	Beta inverse cumulative distribution function
betaln	Logarithm of beta function
ellipj	Jacobi elliptic functions
ellipke	Complete elliptic integrals of first and second kind
erf	Error function
erfc	Complementary error function
erfcinv	Inverse complementary error function
erfcx	Scaled complementary error function
erfinv	Inverse error function
expint	Exponential integral
gamma	Gamma function
gammainc	Incomplete gamma function
gammaincinv	Inverse incomplete gamma function
gammaln	Logarithm of gamma function
legendre	Associated Legendre functions

Sparse Matrices

Elementary Sparse Matrices	Create random and nonrandom sparse matrices
Full to Sparse Conversion	Convert full matrix to sparse, sparse matrix to full
Sparse Matrix Manipulation	Test matrix for sparseness, get information on sparse matrix, allocate sparse matrix, apply function to nonzero elements, visualize sparsity pattern
Reordering Algorithms	Random, column, minimum degree, Dulmage-Mendelsohn, and reverse Cuthill-McKee permutations
Linear Algebra	Compute norms, eigenvalues, factorizations, least squares, structural rank
Linear Equations (Iterative Methods)	Methods for conjugate and biconjugate gradients, residuals, lower quartile
Tree Operations	Elimination trees, tree plotting, factorization analysis

Elementary Sparse Matrices

spdiags	Extract and create sparse band and diagonal matrices
speye	Sparse identity matrix
sprand	Sparse uniformly distributed random matrix
sprandn	Sparse normally distributed random matrix
sprandsym	Sparse symmetric random matrix

Full to Sparse Conversion

find	Find indices and values of nonzero elements
full	Convert sparse matrix to full matrix
sparse	Create sparse matrix
spconvert	Import matrix from sparse matrix external format

Sparse Matrix Manipulation

issparse	Determine whether input is sparse
nnz	Number of nonzero matrix elements
nonzeros	Nonzero matrix elements
nzmax	Amount of storage allocated for nonzero matrix elements
spalloc	Allocate space for sparse matrix
spfun	Apply function to nonzero sparse matrix elements
spones	Replace nonzero sparse matrix elements with ones
spparms	Set parameters for sparse matrix routines
spy	Visualize sparsity pattern

Reordering Algorithms

amd	Approximate minimum degree permutation
colamd	Column approximate minimum degree permutation
colperm	Sparse column permutation based on nonzero count
dmperm	Dulmage-Mendelsohn decomposition
ldl	Block LDL' factorization for Hermitian indefinite matrices
randperm	Random permutation
symamd	Symmetric approximate minimum degree permutation
symrcm	Sparse reverse Cuthill-McKee ordering

Linear Algebra

cholinc	Sparse incomplete Cholesky and Cholesky-Infinity factorizations
condest	1-norm condition number estimate
eigs	Largest eigenvalues and eigenvectors of matrix
ichol	Incomplete Cholesky factorization
ilu	Sparse incomplete LU factorization
luinc	Sparse incomplete LU factorization
normest	2-norm estimate
spaugment	Form least squares augmented system
sprank	Structural rank
svds	Find singular values and vectors

Linear Equations (Iterative Methods)

bicg	Biconjugate gradients method
bicgstab	Biconjugate gradients stabilized method
bicgstabl	Biconjugate gradients stabilized (l) method
cgs	Conjugate gradients squared method
gmres	Generalized minimum residual method (with restarts)
lsqr	LSQR method
minres	Minimum residual method
pcg	Preconditioned conjugate gradients method
qmr	Quasi-minimal residual method
symmlq	Symmetric LQ method
tfqmr	Transpose-free quasi-minimal residual method

Tree Operations

etree	Elimination tree
etreeplot	Plot elimination tree
gplot	Plot nodes and links representing adjacency matrix
symbfact	Symbolic factorization analysis
treelayout	Lay out tree or forest
treeplot	Plot picture of tree
unmesh	Convert edge matrix to coordinate and Laplacian matrices

Math Constants

eps	Floating-point relative accuracy
i	Imaginary unit
Inf	Infinity
intmax	Largest value of specified integer type
intmin	Smallest value of specified integer type
j	Imaginary unit
NaN	Not-a-Number
pi	Ratio of circle's circumference to its diameter
realmax	Largest positive floating-point number
realmin	Smallest positive normalized floating-point number

Data Import and Export

Data Analysis

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