Wikibooks

Julia for MATLAB Users/Core Language/Mathematics

< Julia for MATLAB Users

Mathematics

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Most of the below functionality described in the core MATLAB Mathematics documentation has equivalent, often identical, functionality (more often that not with the same syntax) described in the Base.Mathematics section of the Julia manual. Specific equivalents are identified below; often these have the same names as in Matlab, otherwise the Julia equivalent name is noted.

Elementary Math

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Arithmetic

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See Arithmetic Operators in the Julia manual. Note that in Julia the operators are themselves methods and can be used anywhere a method can. See e.g. the example in the documentation for Base.map.

plus Addition

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uplus Unary plus

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minus Subtraction

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uminus Unary minus

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times Element-wise multiplication

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rdivide Right array division

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ldivide Left array division

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power Element-wise power

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mtimes Matrix Multiplication

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mrdivide Solve systems of linear equations   for  

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mldivide Solve systems of linear equations   for  

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mpower Matrix power

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cumprod Cumulative product

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cumsum Cumulative sum

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diff Differences and Approximate Derivatives

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movsum Moving sum

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prod Product of array elements

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sum Sum of array elements

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ceil Round toward positive infinity

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fix Round toward zero

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floor Round toward negative infinity

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idivide Integer division with rounding option

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mod Remainder after division (modulo operation)

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rem Remainder after division

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round Round to nearest decimal or integer

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bsxfun Apply element-wise operation to two arrays with implicit expansion enabled

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Trigonometry

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See Trigonometric and Hyperbolic functions in the Julia manual.

sin Sine of argument in radians

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sind Sine of argument in degrees

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asin Inverse sine in radians

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asind Inverse sine in degrees

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sinh Hyperbolic sine of argument in radians

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asinh Inverse hyperbolic sine

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cos Cosine of argument in radians

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cosd Cosine of argument in degrees

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acos Inverse cosine in radians

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acosd Inverse cosine in degrees

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cosh Hyperbolic cosine

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acosh Inverse hyperbolic cosine

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tan Tangent of argument in radians

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tand Tangent of argument in degrees

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atan Inverse tangent in radians

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atand Inverse tangent in degrees

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atan2 Four-quadrant inverse tangent

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atan2d Four-quadrant inverse tangent in degrees

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tanh Hyperbolic tangent

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atanh Inverse hyperbolic tangent

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csc Cosecant of input angle in radians

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cscd Cosecant of argument in degrees

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acsc Inverse cosecant in radians

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acscd Inverse cosecant in degrees

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csch Hyperbolic cosecant

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acsch Inverse hyperbolic cosecant

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sec Secant of angle in radians

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secd Secant of argument in degrees

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asec Inverse secant in radians

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asecd Inverse secant in degrees

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sech Hyperbolic secant

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asech Inverse hyperbolic secant

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cot Cotangent of angle in radians

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cotd Cotangent of argument in degrees

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acot Inverse cotangent in radians

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acotd Inverse cotangent in degrees

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coth Hyperbolic cotangent

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acoth Inverse hyperbolic cotangent

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hypot Square root of sum of squares (hypotenuse)

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deg2rad Convert angle from degrees to radians

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rad2deg Convert angle from radians to degrees

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Exponents and Logarithms

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See Powers, logs and roots in the Julia manual.

exp Exponential

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expm1 Compute   accurately for small values of x

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log Natural logarithm

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log10 Common logarithm (base 10)

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log1p Compute log(1+x) accurately for small values of x

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log2 Base 2 logarithm and floating-point number dissection

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nextpow2 Exponent of next higher power of 2

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nthroot Real nth root of real numbers

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pow2 Base 2 power and scale floating-point numbers

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reallog Natural logarithm for nonnegative real arrays

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realpow Array power for real-only output

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realsqrt Square root for nonnegative real arrays

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sqrt Square root

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Complex Numbers

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See Complex Numbers in the Julia manual.

abs Absolute value and complex magnitude

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angle Phase angle

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complex Create complex array

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conj Complex conjugate

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cplxpair Sort complex numbers into complex conjugate pairs

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i Imaginary unit

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imag Imaginary part of complex number

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isreal Determine whether array is real

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j Imaginary unit

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real Real part of complex number

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sign Sign function (signum function)

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unwrap Correct phase angles to produce smoother phase plots

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Discrete Math

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Equivalents available in Julia Base

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factorial Factorial of input
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gcd Greatest common divisor
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lcm Least common multiple
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Equivalents available in JuliaMath/Primes.jl

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factor Prime factors
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primes Prime numbers less than or equal to input value
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isprime Determine which array elements are prime
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nchoosek Binomial coefficient or all combinations (Julia: binomial)
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Others

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perms All possible permutations
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The Julia Permutations.permutations(a) function ( Permutations.jl package) returns an iterator object (because the number of permutations can be very large), and in lexicographic order rather than reverse lexicographic. Therefore a drop-in equivalent could be constructed as follows:

julia> perms(a) = reverse(collect(permutations(a)))
perms (generic function with 1 method)

julia> perms([2,4,6])
6-element Array{Array{Int64,1},1}:
 [6, 4, 2]
 [6, 2, 4]
 [4, 6, 2]
 [4, 2, 6]
 [2, 6, 4]
 [2, 4, 6]
rat Rational fraction approximation, rats Rational output
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There doesn't appear to be a direct Julia equivalent of these, but note that unlike Matlab, Julia has a native Rational Number type .

Polynomials

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See the Polynomials.jl package. Note that this package provides a proper type for polynomials, Polynomials.Poly, while in Matlab a polynomial of degree  is represented by a vector of length  whose elements are the coefficients in descending powers of the polynomial.

poly Polynomial with specified roots or characteristic polynomial

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polyeig Polynomial eigenvalue problem

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polyfit Polynomial curve fitting

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Polynomials.polyfit provides comparable basic functionality--the single output argument form of the Matlab function--although it lacks the additional error estimate and centering/scaling features.

residue Partial fraction expansion (partial fraction decomposition)

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roots Polynomial roots

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Polynomials.roots provides roots with multiplicity.

polyval Polynomial evaluation

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See Base.Math.@evalpoly in the Julia Manual.

polyvalm Matrix polynomial evaluation

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conv Convolution and polynomial multiplication

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deconv Deconvolution and polynomial division

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polyint Polynomial integration

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polyder Polynomial differentiation

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Special Functions

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airy Airy Functions

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besselh Bessel function of third kind (Hankel function)

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besseli Modified Bessel function of first kind

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besselj Bessel function of first kind

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besselk Modified Bessel function of second kind

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bessely Bessel function of second kind

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beta Beta function

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betainc Incomplete beta function

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betaincinv Beta inverse cumulative distribution function

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betaln Logarithm of beta function

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ellipj Jacobi elliptic functions

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ellipke Complete elliptic integrals of first and second kind

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erf Error function

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erfc Complementary error function

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erfcinv Inverse complementary error function

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erfcx Scaled complementary error function

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erfinv Inverse error function

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expint Exponential integral

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gamma Gamma function

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gammainc Incomplete gamma function

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gammaincinv Inverse incomplete gamma function

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gammaln Logarithm of gamma function

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legendre Associated Legendre functions

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psi Psi (polygamma) function

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Cartesian Coordinate System Conversion

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cart2pol Transform Cartesian coordinates to polar or cylindrical

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cart2sph Transform Cartesian coordinates to spherical

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pol2cart Transform polar or cylindrical coordinates to Cartesian

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sph2cart Transform spherical coordinates to Cartesian

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Constants and Test Matrices

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Constants

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See General Number Functions and Constants in the Julia manual.

eps Floating-point relative accuracy

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flintmax Largest consecutive integer in floating-point format

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i, j Imaginary unit

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In Julia, im is equivalent; this allows i and j to be used as e.g. loop indices without conflict.

Inf Infinity

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pi Ratio of circle's circumference to its diameter

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Also available as pi in Julia as well as \piTab ↹ →  

NaN Not-a-Number

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isfinite Array elements that are finite

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isinf Array elements that are infinite

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isnan Array elements that are NaN

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compan Companion matrix

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Test Matrices

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See the MatrixDepot.jl package; most of the matrices in gallery and all the rest below are available in that package, plus some additional ones.

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hadamard Hadamard matrix

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hankel Hankel matrix

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hilb Hilbert matrix

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invhilb Inverse of Hilbert matrix

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magic Magic square

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pascal Pascal matrix

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rosser Classic symmetric eigenvalue test problem

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toeplitz Toeplitz matrix

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vander Vandermonde matrix

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wilkinson Wilkinson's eigenvalue test matrix

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Linear Algebra

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See Linear Algebra in the Julia manual.

mldivide Solve systems of linear equations   for  

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mrdivide Solve systems of linear equations   for  

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decomposition Matrix decomposition for solving linear systems

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lsqminnorm Minimum norm least-squares solution to linear equation

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linsolve Solve linear system of equations

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inv Matrix inverse

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pinv Moore-Penrose pseudoinverse

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lscov Least-squares solution in presence of known covariance

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lsqnonneg Solve nonnegative linear least-squares problem

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sylvester Solve Sylvester equation   for  

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eig Eigenvalues and eigenvectors

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eigs Subset of eigenvalues and eigenvectors

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balance Diagonal scaling to improve eigenvalue accuracy

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svd Singular value decomposition

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svds Subset of singular values and vectors

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gsvd Generalized singular value decomposition

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ordeig Eigenvalues of quasitriangular matrices

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ordqz Reorder eigenvalues in QZ factorization

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ordschur Reorder eigenvalues in Schur factorization

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polyeig Polynomial eigenvalue problem

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qz QZ factorization for generalized eigenvalues

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hess Hessenberg form of matrix

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schur Schur decomposition

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rsf2csf Convert real Schur form to complex Schur form

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cdf2rdf Convert complex diagonal form to real block diagonal form

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lu LU matrix factorization

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ldl Block LDL' factorization for Hermitian indefinite matrices

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chol Cholesky factorization

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cholupdate Rank 1 update to Cholesky factorization

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qr Orthogonal-triangular decomposition

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qrdelete Remove column or row from QR factorization

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qrinsert Insert column or row into QR factorization

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qrupdate Rank 1 update to QR factorization

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planerot Givens plane rotation

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transpose Transpose vector or matrix

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ctranspose Complex conjugate transpose

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mtimes Matrix Multiplication

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mpower Matrix power

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sqrtm Matrix square root

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expm Matrix exponential

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logm Matrix logarithm

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funm Evaluate general matrix function

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kron Kronecker tensor product

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cross Cross product

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dot Dot product

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bandwidth Lower and upper matrix bandwidth

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tril Lower triangular part of matrix

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triu Upper triangular part of matrix

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isbanded Determine if matrix is within specific bandwidth

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isdiag Determine if matrix is diagonal

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ishermitian Determine if matrix is Hermitian or skew-Hermitian

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issymmetric Determine if matrix is symmetric or skew-symmetric

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istril Determine if matrix is lower triangular

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istriu Determine if matrix is upper triangular

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norm Vector and matrix norms

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normest 2-norm estimate

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vecnorm Vector-wise norm

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cond Condition number for inversion

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condest 1-norm condition number estimate

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rcond Reciprocal condition number

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condeig Condition number with respect to eigenvalues

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det Matrix determinant

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null Null space

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orth Orthonormal basis for range of matrix

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rank Rank of matrix

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rref Reduced row echelon form (Gauss-Jordan elimination)

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trace Sum of diagonal elements

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subspace Angle between two subspaces

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Random Number Generation

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rand Uniformly distributed random numbers

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randn Normally distributed random numbers

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randi Uniformly distributed pseudorandom integers

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randperm Random permutation

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rng Control random number generation

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randStream Random number stream

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Interpolation

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interp1 1-D data interpolation (table lookup)

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interp2 Interpolation for 2-D gridded data in meshgrid format

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interp3 Interpolation for 3-D gridded data in meshgrid format

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interpn Interpolation for 1-D, 2-D, 3-D, and N-D gridded data in ndgrid format

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gRiddedinterpolant Gridded data interpolation

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pchip Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)

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spline Cubic spline data interpolation

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ppval Evaluate piecewise polynomial

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mkpp Make piecewise polynomial

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unmkpp Extract piecewise polynomial details

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padecoef Padé approximation of time delays

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interpft 1-D interpolation (FFT method)

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ndgrid Rectangular grid in N-D space

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meshgrid 2-D and 3-D grids

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griddata Interpolate 2-D or 3-D scattered data

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griddatan Interpolate N-D scattered data

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scaTteredinterpolant Interpolate 2-D or 3-D scattered data

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Optimization

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fminbnd Find minimum of single-variable function on fixed interval

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fminsearch Find minimum of unconstrained multivariable function using derivative-free method

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lsqnonneg Solve nonnegative linear least-squares problem

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fzero Root of nonlinear function

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optimget Optimization options values

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optimset Create or edit optimization options structure

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Numerical Integration and Differential Equations

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See DifferentialEquations.jl. In particular see the section Translations from MATLAB/Python/R.

Ordinary Differential Equations

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ode45 Solve nonstiff differential equations — medium order method

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ode23 Solve nonstiff differential equations — low order method

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ode113 Solve nonstiff differential equations — variable order method

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ode15s Solve stiff differential equations and DAEs — variable order method

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ode23s Solve stiff differential equations — low order method

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ode23t Solve moderately stiff ODEs and DAEs — trapezoidal rule

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ode23tb Solve stiff differential equations — trapezoidal rule + backward differentiation formula

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ode15i Solve fully implicit differential equations — variable order method

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decic Compute consistent initial conditions for ode15i

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odeget Extract ODE option values

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odeset Create or modify options structure for ODE solvers

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deval Evaluate differential equation solution structure

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odextend Extend solution to ODE

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Boundary Value Problems

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bvp4c Solve boundary value problems for ordinary differential equations

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bvp5c Solve boundary value problems for ordinary differential equations

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bvpinit Form initial guess for BVP solvers

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bvpxtend Form guess structure for extending boundary value solutions

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bvpget Extract properties from options structure created with bvpset

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bvpset Create or alter options structure of boundary value problem

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deval Evaluate differential equation solution structure

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Delay Differential Equations

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dde23 Solve delay differential equations (DDEs) with constant delays

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ddesd Solve delay differential equations (DDEs) with general delays

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ddensd Solve delay differential equations (DDEs) of neutral type

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ddeget Extract properties from delay differential equations options structure

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ddeset Create or alter delay differential equations options structure

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deval Evaluate differential equation solution structure

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Partial Differential Equations

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pdepe Solve initial-boundary value problems for parabolic-elliptic PDEs in 1-D

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pdeval Evaluate numerical solution of PDE using output of pdepe

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Numerical Integration and Differentiation

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integral Numerical integration

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integral2 Numerically evaluate double integral

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integral3 Numerically evaluate triple integral

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quadgk Numerically evaluate integral, adaptive Gauss-Kronrod quadrature

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quad2d Numerically evaluate double integral, tiled method

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cumtrapz Cumulative trapezoidal numerical integration

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trapz Trapezoidal numerical integration

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polyint Polynomial integration

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del2 Discrete Laplacian

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diff Differences and Approximate Derivatives

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gradient Numerical gradient

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polyder Polynomial differentiation

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Fourier Analysis and Filtering

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fft Fast Fourier transform

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fft2 2-D fast Fourier transform

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fftn N-D fast Fourier transform

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fftshift Shift zero-frequency component to center of spectrum

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fftw Define method for determining FFT algorithm

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ifft Inverse fast Fourier transform

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ifft2 2-D inverse fast Fourier transform

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ifftn Multidimensional inverse fast Fourier transform

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ifftshift Inverse zero-frequency shift

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nextpow2 Exponent of next higher power of 2

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interpft 1-D interpolation (FFT method)

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conv Convolution and polynomial multiplication

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conv2 2-D convolution

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convn N-D convolution

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deconv Deconvolution and polynomial division

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filter 1-D digital filter

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filter2 2-D digital filter

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ss2tf Convert state-space representation to transfer function

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padecoef Padé approximation of time delays

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Sparse Matrices

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spalloc Allocate space for sparse matrix

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spdiags Extract and create sparse band and diagonal matrices

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speye Sparse identity matrix

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sprand Sparse uniformly distributed random matrix

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sprandn Sparse normally distributed random matrix

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sprandsym Sparse symmetric random matrix

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sparse Create sparse matrix

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spconvert Import from sparse matrix external format

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issparse Determine whether input is sparse

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nnz Number of nonzero matrix elements

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nonzeros Nonzero matrix elements

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nzmax Amount of storage allocated for nonzero matrix elements

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spfun Apply function to nonzero sparse matrix elements

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spones Replace nonzero sparse matrix elements with ones

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spparms Set parameters for sparse matrix routines

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spy Visualize sparsity pattern

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find Find indices and values of nonzero elements

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full Convert sparse matrix to full matrix

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dissect Nested dissection permutation

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amd Approximate minimum degree permutation

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colamd Column approximate minimum degree permutation

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colperm Sparse column permutation based on nonzero count

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dmperm Dulmage-Mendelsohn decomposition

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randperm Random permutation

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symamd Symmetric approximate minimum degree permutation

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symrcm Sparse reverse Cuthill-McKee ordering

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pcg Preconditioned conjugate gradients method

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minres Minimum residual method

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symmlq Symmetric LQ method

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gmres Generalized minimum residual method (with restarts)

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bicg Biconjugate gradients method

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bicgstab Biconjugate gradients stabilized method

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bicgstabl Biconjugate gradients stabilized (l) method

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cgs Conjugate gradients squared method

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qmr Quasi-minimal residual method

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tfqmr Transpose-free quasi-minimal residual method

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lsqr LSQR method

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ichol Incomplete Cholesky factorization

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ilu Incomplete LU factorization

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eigs Subset of eigenvalues and eigenvectors

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svds Subset of singular values and vectors

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normest 2-norm estimate

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condest 1-norm condition number estimate

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sprank Structural rank

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etree Elimination tree

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symbfact Symbolic factorization analysis

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spaugment Form least-squares augmented system

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dmperm Dulmage-Mendelsohn decomposition

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etreeplot Plot elimination tree

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treelayout Lay out tree or forest

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treeplot Plot picture of tree

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unmesh Convert edge matrix to coordinate and Laplacian matrices

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Graph and Network Algorithms

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graph Graph with undirected edges

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digraph Graph with directed edges

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addnode Add new node to graph

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rmnode Remove node from graph

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addedge Add new edge to graph

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rmedge Remove edge from graph

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flipedge Reverse edge directions

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numnodes Number of nodes in graph

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numedges Number of edges in graph

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findnode Locate node in graph

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findedge Locate edge in graph

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edgecount Number of edges between two nodes

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reordernodes Reorder graph nodes

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subgraph Extract subgraph

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centrality Measure node importance

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maxflow Maximum flow in graph

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conncomp Connected graph components

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biconncomp Biconnected graph components

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condensation Graph condensation

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bctree Block-cut tree graph

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minspantree Minimum spanning tree of graph

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toposort Topological order of directed acyclic graph

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isdag Determine if graph is acyclic

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transclosure Transitive closure

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transreduction Transitive reduction

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isisomorphic Determine whether two graphs are isomorphic

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isomorphism Compute isomorphism between two graphs

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ismultigraph Determine whether graph has multiple edges

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simplify Reduce multigraph to simple graph

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shortestpath Shortest path between two single nodes

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shortestpathtree Shortest path tree from node

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distances Shortest path distances of all node pairs

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adjacency Graph adjacency matrix

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incidence Graph incidence matrix

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laplacian Graph Laplacian matrix

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degree Degree of graph nodes

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neighbors Neighbors of graph node

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nearest Nearest neighbors within radius

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indegree In-degree of nodes

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outdegree Out-degree of nodes

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predecessors Node predecessors

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successors Node successors

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inedges Incoming edges to node

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outedges Outgoing edges from node

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plot Plot graph nodes and edges

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labeledge Label graph edges

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labelnode Label graph nodes

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layout Change layout of graph plot

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highlight Highlight nodes and edges in plotted graph

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graphPlot Graph plot for directed and undirected graphs

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Computational Geometry

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See the JuliaGeometry GitHub organization.

Triangulation Representation

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triangulation Triangulation in 2-D or 3-D

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tetramesh Tetrahedron mesh plot

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trimesh Triangular mesh plot

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triplot 2-D triangular plot

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trisurf Triangular surface plot

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Delaunay Triangulation

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deLaunaytriangulation Delaunay triangulation in 2-D and 3-D

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delaunay Delaunay triangulation

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delaunayn N-D Delaunay triangulation

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tetramesh Tetrahedron mesh plot

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trimesh Triangular mesh plot

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triplot 2-D triangular plot

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trisurf Triangular surface plot

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triangulation Triangulation in 2-D or 3-D

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deLaunaytriangulation Delaunay triangulation in 2-D and 3-D

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delaunay Delaunay triangulation

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delaunayn N-D Delaunay triangulation

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Bounding Regions

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boundary Boundary of a set of points in 2-D or 3-D

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alphaShape Polygons and polyhedra from points in 2-D and 3-D

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convhull Convex hull

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convhulln N-D convex hull

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Voronoi Diagram

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patch Create one or more filled polygons

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voronoi Voronoi diagram

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voronoin N-D Voronoi diagram

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Elementary Polygons

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The Julia package GeometricalPredicates.jl provides some similar functionality.

inpolygon Points located inside or on edge of polygonal region

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nsidedpoly Regular polygon

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polyarea Area of polygon

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polybuffer Create buffer around points or lines

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rectint Rectangle intersection area

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polyshape 2-D polygons

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addboundary Add polyshape boundary

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rmboundary Remove polyshape boundary

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rmholes Remove holes in polyshape

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rmslivers Remove polyshape boundary outliers

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simplify Simplify polyshape boundaries

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boundary Vertex coordinates of polyshape boundary

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isequal Determine if polyshape objects are equal

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ishole Determine if polyshape boundary is a hole

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isinterior Query points inside polyshape

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issimplified Determine if polyshape is well-defined

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nearestvertex Query nearest polyshape vertex

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numboundaries Number of polyshape boundaries

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numsides Number of polyshape sides

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overlaps Determine whether polyshape objects overlap

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area Area of polyshape

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boundingbox Bounding box of polyshape

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centroid Centroid of polyshape

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convhull Convex hull of polyshape

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perimeter Perimeter of polyshape

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triangulation Triangulate polyshape

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turningdist Compute turning distance between polyshape objects

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intersect Intersection of polyshape objects

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subtract Difference of two polyshape objects

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union Union of polyshape objects

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xor Exclusive OR of two polyshape objects

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polybuffer Buffer polyshape

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rotate Rotate polyshape

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scale Scale polyshape

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translate Translate polyshape

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holes Convert polyshape hole boundaries to array of polyshape objects

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plot Plot polyshape

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regions Access polyshape regions

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sortboundaries Sort polyshape boundaries

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sortregions Sort polyshape regions

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