Function Reference

The Multiprecision Computing Toolbox extends MATLAB with a wide set of mathematical functions capable to calculate results with arbitrary precision. The toolbox supplies multiprecision equivalents to the majority of the standard MATLAB routines, and also adds a few of its own.

Every function in toolbox is implemented in multiple different modifications in order to support all special cases and features of MATLAB’s standard routines.

Thus there is no syntactic difference in calling high precision functions compared to default double routines. Exactly the same code can be used, only input arguments should be changed to multiprecision type.

Information on toolbox-specific routines can be obtained by the MATLAB help command (e.g. help mp.Digits, help mp.GaussLegendre, etc.)

For example, arbitrary precision ifft() and svd() functions cover all the following special cases:

y = ifft(X)
y = ifft(X,n)
y = ifft(X,[],dim)
y = ifft(X,n,dim)
y = ifft(..., 'symmetric')
y = ifft(..., 'nonsymmetric')
s = svd(X)
[U,S,V] = svd(X)
[U,S,V] = svd(X,0)
[U,S,V] = svd(X,'econ')

This compatibility allows the user to effortlessly port existing programs to the arbitrary precision computing provided by the toolbox (see examples of Arbitrary precision Gauss-Legendre Quadrature, Krylov Subspace Methods in Arbitrary Precision, Gauss-Kronrod Quadrature).

This page provides a list of all the supported functions available to the user.

In addition, the Multiprecision Computing Toolbox is under active development with frequent updates to functionality (new features, improved performance) and reference materials. To automatically receive notifications about any relevant updates, please subscribe to the RSS feed found in the top right corner of the page.

Multiprecision Toolbox Specifics

User’s Manual and Blog Articles demonstrate the usage of the functions listed below.

Precision Control

mp.Digits([digits])Setup the default accuracy. All subsequent multiprecision calculations will be conducted using the newly specified number of decimal digits. The function returns the current default precision if no digits are supplied.
mp.GuardDigits([digits])Setup the number of guard digits. Internally, toolbox uses mp.Digits() + mp.GuardDigits() for precision in all operations.

Expression Evaluation

mp(expression [,digits])Evaluate an expression with the accuracy of decimal digits. Complex and real arithmetic is supported, including constants, trigonometric, exponential, and special functions.

mp('[sqrt(-1) pi^i; exp(1/16) sin(pi/4)]')

The default precision (defined by mp.Digits()) is used if no digits are supplied.

Type Conversion

mp(entity [,digits])Converts MATLAB’s existing numbers and matrices into arbitrary precision.


Please note that it is better to avoid the direct conversion of MATLAB’s floating-point arrays into multiprecision form. Instead, re-calculate them using the required extended precision from the onset:


For further information and help, please check the Existing Code Porting and How to write precision independent code in MATLAB.

Diagnostics & Tests

mp.InfoShows information on the toolbox, including version, license, and its major components.
mp.TestRuns a comprehensive test of all major functions in the toolbox.

Arrays and Matrices

Arithmetic Operators

A+B Addition
A-B Subtraction
A*B Matrix multiplication
A^B Matrix power
A\B Backslash or left matrix divide
A/B Slash or right matrix divide
A' Transpose
A.' Nonconjugated transpose
A.*B Array multiplication (element-wise)
A.^B Array power (element-wise)
A.\B Left array divide (element-wise)
A./B Right array divide (element-wise)
+A Unary plus
-B Unary minus
ctranspose Complex conjugate transpose
minus Minus
mpower Matrix power
plus Plus
times Array multiply
transpose Transpose
uminus Unary minus
uplus Unary plus

Elementary Matrices and Arrays

accumarrayConstruct array with accumulation
blkdiag Construct block diagonal matrix from input arguments
colon (:)Create vectors, array subscripting, and for-loop iterators
linspaceGenerate linearly spaced vectors
logspaceGenerate logarithmically spaced vectors
meshgridRectangular grid in 2-D and 3-D space
ndgridRectangular grid in N-D space
diag Diagonal matrices and diagonals of matrix
eye(...,'mp') Identity matrix
ones(...,'mp')Create array of all ones
zeros(...,'mp')Create array of all zeros
rand(...,'mp') Uniformly distributed pseudorandom numbers
randn(...,'mp')Normally distributed pseudorandom numbers

Array Operations

cast Cast variable to different data type
cross Vector cross product
cumprod Cumulative product
cumsum Cumulative sum
diff Differences and approximate derivatives
dot Vector dot product
kronKronecker tensor product
prod Product of array elements
sum Sum of array elements
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
circshiftShift array circularly
diag Diagonal matrices and diagonals of matrix
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
ipermuteInverse 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
shiftdimShift dimensions
sort Sort array elements in ascending or descending order
squeeze Remove singleton dimensions
vertcat Concatenate arrays vertically

Specialized Matrices

compan Companion matrix
hankel Hankel matrix
toeplitz Toeplitz matrix
vander Vandermonde matrix
hilb(n,'mp') Hilbert matrix
invhilb(n,'mp')Inverse of Hilbert matrix
eye(...,'mp') Identity matrix
ones(...,'mp')Create array of all ones
zeros(...,'mp')Create array of all zeros
wilkinson(n,'mp') Wilkinson’s eigenvalue test matrix
hadamard(n,'mp') Hadamard matrix
magic(n,'mp') Magic square
pascal(n,m,'mp') Pascal matrix
rosser('mp') Classic symmetric eigenvalue test problem

Basic Information

dispDisplay text or array
displayDisplay text or array (overloaded method)
lengthLength of vector or largest array dimension
maxLargest elements in array
minSmallest elements in array
findFind indices and values of nonzero elements
ndimsNumber of array dimensions
numelNumber of elements in array or subscripted array expression
sizeArray dimensions
iscolumnDetermine 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
isfiniteArray elements that are finite
isfloat Determine whether input is floating-point array
isinfArray elements that are infinite
isintegerDetermine whether input is integer array
islogicalDetermine whether input is logical array
ismatrixDetermine whether input is matrix
isnanArray elements that are NaN
isnumericDetermine whether input is numeric array
ismpDetermine whether input is multiprecision entity
isrowDetermine whether input is row vector
isscalarDetermine whether input is scalar
issparseDetermine whether input is sparse
isvectorDetermine whether input is vector
hasInfNaNDetermine whether input has Inf or NaN entries.

Elementary Math


acos Inverse cosine
acosh Inverse hyperbolic cosine
acot Inverse cotangent
acoth Inverse hyperbolic cotangent
acsc Inverse cosecant
acsch Inverse hyperbolic cosecant
asec Inverse secant
asech Inverse hyperbolic secant
asin Inverse sine
asinh Inverse hyperbolic sine
atan Inverse tangent
atan2 Four-quadrant inverse tangent
atanh Inverse hyperbolic tangent
cos Cosine of argument
cosh Hyperbolic cosine
cot Cotangent of argument
coth Hyperbolic cotangent
csc Cosecant of argument
csch Hyperbolic cosecant
hypot Square root of sum of squares
sec Secant of argument
sech Hyperbolic secant
sin Sine of argument
sinh Hyperbolic sine of argument
tan Tangent of argument
tanh Hyperbolic tangent


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
nextpow2Next 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
realsqrtSquare root for nonnegative real arrays
sqrt Square root


abs Absolute value and complex magnitude
angle Phase angle
complex Construct complex data from real and imaginary components
conj Complex conjugate
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

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

factorPrime factors
factorialFactorial function
gcdGreatest common divisor
isprimeDetermine which array elements are prime
lcmLeast common multiple
primesPrime numbers less than or equal to input value
nextprimeNext prime greater than input value
prevprimePrevious prime less than input value

Linear Algebra

Matrix Operations

cross Cross product
dot Dot product
kron Kronecker tensor product
tril Lower triangular part of matrix
triu Upper triangular part of matrix
transpose Transpose vector or matrix

Linear Equations

cond Condition number with respect to inversion
inv Matrix inverse
linsolveSolve linear system of equations
pinv Moore-Penrose pseudoinverse of matrix
rcond Matrix reciprocal condition number estimate
sylvesterSolve Sylvester equation AX + XB = C for X
sylvester_triSolve (quasi-)triangular Sylvester equations
lyapSolve continuous Lyapunov equations
dlyapSolve discrete-time Lyapunov equations
mldivide Solve systems of linear equations Ax = B for x
mrdivide Solve systems of linear equations xA = B for x

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)
minresMinimum residual method
pcg Preconditioned conjugate gradients method

Matrix Decomposition

chol Cholesky factorization
cholupdateRank 1 update to Cholesky factorization
lu LU matrix factorization
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
planerot Givens plane rotation
ldl Block LDL’ factorization for Hermitian indefinite matrices
cdf2rdf Convert complex diagonal form to real block diagonal form
rsf2csf Convert real Schur form to complex Schur form
svd Singular value decomposition
gsvd Generalized singular value decomposition
schur Schur decomposition
qz Generalized Schur decomposition

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/mpeigs Subset of eigenvalues and eigenvectors
hess Hessenberg form of matrix
ordeig Eigenvalues of quasi-triangular matrices
ordqz Reorder eigenvalues in QZ factorization
ordschurReorder eigenvalues in Schur factorization
poly Polynomial with specified roots
polyeig Polynomial eigenvalue problem
qz QZ factorization for generalized eigenvalues
rsf2csf Convert real Schur form to complex Schur form
schur Schur decomposition
isschur Determine whether a matrix is in Schur form (conformed with LAPACK)
sqrtm Matrix square root
svd Singular value decomposition
gsvd Generalized singular value decomposition

Matrix Analysis

bandwidth Lower and upper matrix bandwidth
cond Condition number with respect to inversion
det Matrix determinant
isbanded Determine if matrix is within specific bandwidth
isdiag Determine whether a matrix is diagonal
ishermitianDetermine whether a matrix is real symmetric or complex Hermitian
issymmetricDetermine whether a matrix is real or complex symmetric
istril Determine whether a matrix is lower triangular
istriu Determine whether a matrix is upper triangular
isschur Determine whether a matrix is in Schur form (conformed with LAPACK)
norm Vector and matrix norms
null Null space of a matrix
orth Orthonormal basis for range of matrix
rank Rank of matrix
rcond Matrix reciprocal condition number estimate
rref Reduced row echelon form (Gauss-Jordan elimination)
subspace Angle between two subspaces
trace Sum of diagonal elements

Matrix Functions

expm Matrix exponential
logm Matrix logarithm
sqrtm Matrix square root
bsxfun Apply element-by-element binary operation to two arrays with singleton expansion enabled
funm Evaluate general matrix function
arrayfunApply function to each element of array
mpower Matrix power
sinm Matrix sine
cosm Matrix cosine
sinhm Matrix hyperbolic sine
coshm Matrix hyperbolic cosine
sqrtm_triSquare root of upper triangular matrix

Numerical Methods

Numerical Integration and Differentiation

integralNumerical integration
integral2Numerically evaluate double integral
integral3Numerically evaluate triple integral
quadgkNumerically evaluate integral, adaptive Gauss-Kronrod quadrature
quad2dNumerically evaluate double integral, tiled method
cumtrapzCumulative trapezoidal numerical integration
trapzTrapezoidal numerical integration
polyintPolynomial integration
quadNumerically evaluate integral, adaptive Simpson quadrature
quadglNumerically evaluate integral, fixed Gauss-Legendre quadrature
dblquadNumerically evaluate double integral over rectangle
triplequadNumerically evaluate triple integral
mp.GaussKronrodCompute Gauss-Kronrod quadrature’s abscissae and weights
mp.GaussLegendreCompute Gauss-Legendre quadrature’s abscissae and weights
mp.GaussChebyshevType1Compute Gauss-Chebyshev Type 1 quadrature’s abscissae and weights
mp.GaussChebyshevType2Compute Gauss-Chebyshev Type 2 quadrature’s abscissae and weights
mp.GaussGegenbauerCompute Gauss-Gegenbauer quadrature’s abscissae and weights
mp.GaussJacobiCompute Gauss-Jacobi quadrature’s abscissae and weights
mp.GaussLaguerreCompute Gauss-Laguerre quadrature’s abscissae and weights
mp.GaussHermitCompute Gauss-Hermit quadrature’s abscissae and weights
del2 Discrete Laplacian
diff Differences and approximate derivatives
gradientNumerical gradient
polyderPolynomial differentiation


fminsearchFind minimum of unconstrained multivariable function (Nelder-Mead)
fzeroFind root of continuous function of one variable
fsolveSolve system of nonlinear equations
optimsetCreate or edit optimization options structure
optimgetOptimization options values

Ordinary Differential Equations

ode45Solve initial value problems for ordinary differential equations
ode15sSolve stiff differential equations and DAEs; variable order method
ode113Solve nonstiff differential equations with variable order method
odegetOrdinary differential equation options parameters
odesetCreate or alter options structure for ordinary differential equation solvers


polyPolynomial with specified roots
polyderPolynomial derivative
polyeigPolynomial eigenvalue problem
polyfitPolynomial curve fitting
polyintIntegrate polynomial analytically
polyvalPolynomial evaluation
polyvalmMatrix polynomial evaluation
residueConvert between partial fraction expansion and polynomial coefficients
rootsPolynomial roots

Orthogonal Polynomials

chebyshevT Chebyshev polynomials of the first kind
chebyshevU Chebyshev polynomials of the second kind
hermiteH Hermite polynomials
legendreP Legendre polynomials

Data Analysis

Basic Operations

cumprod Cumulative product
cumsum Cumulative sum
prod Product of array elements
sort Sort array elements in ascending or descending order
sum Sum of array elements

Descriptive Statistics

max Largest elements in array
meanAverage or mean value of array
min Smallest elements in array
std Standard deviation

Fourier Transforms

fft Fast Fourier transform
fft2 2-D fast Fourier transform
fftn N-D fast Fourier transform
fftshiftShift zero-frequency component to center of spectrum
ifftInverse fast Fourier transform
ifft22-D inverse fast Fourier transform
ifftnN-D inverse fast Fourier transform
ifftshiftInverse FFT shift

Control Theory

Matrix Computations

lyap Continuous Lyapunov equation solution
dlyap Solve discrete-time Lyapunov equations
care Continuous-time algebraic Riccati equation solution
dare Solve discrete-time algebraic Riccati equations (DAREs)
gcare Generalized solver for continuous-time algebraic Riccati equation
gdare Generalized solver for discrete-time algebraic Riccati equation


1D Interpolation

interp1 1-D data interpolation (table lookup)
pchip Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)
spline Cubic spline data interpolation
ppval Evaluate piecewise polynomial
mkpp Make piecewise polynomial
unmkpp Piecewise polynomial details

Programming and Data Types

Data Type Conversion

num2str Convert number to string
cast Cast variable to different data type
cell2matConvert cell array to ordinary array of the underlying data type
mp.cellfunApply function to each cell in cell array
double Convert to double precision
int16 Convert to 16-bit signed integer
int32 Convert to 32-bit signed integer
int64 Convert to 64-bit signed integer
int8 Convert to 8-bit signed integer
single Convert to single precision
uint16 Convert to 16-bit unsigned integer
uint32 Convert to 32-bit unsigned integer
uint64 Convert to 64-bit unsigned integer
uint8 Convert to 8-bit unsigned integer
logical Convert numeric values to logical
sym2mpConvert ‘sym’ object to multiprecision type.
mp2symConvert multiprecision object to ‘sym’ type.

Logical Operations

allDetermine whether all array elements are nonzero or true
andFind logical AND of array or scalar inputs
anyDetermine whether any array elements are nonzero
isaDetermine whether input is object of given class
notFind logical NOT of array or scalar input
orFind logical OR of array or scalar inputs
xorLogical exclusive-OR
logical Convert numeric values to logical
islogicalDetermine whether input is logical array

Relational Operations

eq, ==Test for equality
ge, >=Test for greater than or equal to
gt, >Test for greater than
le, <=Test for less than or equal to
lt, <Test for less than
ne, ~=Test for inequality

File I/O

fprintfWrite data to text file
mp.writeWrite matrix to specified text file. Use doc mp.write for more information
mp.readLoad matrix from text file. Use doc mp.write for more information

Special Functions

Gamma and related functions

gamma Gamma function
gammaln Logarithm of gamma function
gammaincIncomplete gamma function
psi Digamma function

Error function and Fresnel integrals

erf Error function
erfc Complementary error function
erfi Imaginary error function
FresnelSFresnel sine integral
FresnelCFresnel cosine integral

Lambert W function

lambertwLambert W function (real arguments only)

Bessel functions

besselhBessel function of third kind (Hankel function)
besseliModified Bessel function of first kind
besseljBessel function of first kind
besselkModified Bessel function of second kind
besselyBessel function of second kind

Beta functions

beta Beta function
betaln Logarithm of beta function

Hypergeometric functions

hypergeomGeneralized hypergeometric function: 0F0, 1F0, 0F1, 1F1
KummerM Confluent hypergeometric function of the first kind, M(a,b,z)
KummerU Confluent hypergeometric function of the second kind, U(a,b,z)

Exponential integrals and related functions

expintExponential Integral, \mathop{E_{n}\/}
eint Exponential Integral (in terms of Cauchy principal value), \mathop{\mathrm{Ei}\/}
logintLogarithmic Integral
cosintCosine Integral
sinintSine Integral
chintHyperbolic Cosine Integral
shintHyperbolic Sine Integral

Zeta functions

zeta Riemann zeta function (real arguments only)
LerchPhiLerch transcendent, Φ(z,s,a), for |z|<1


airyAiry Functions
ellipjJacobi elliptic functions
ellipkeComplete elliptic integrals of first and second kind
legendreAssociated Legendre functions
mp.BernoulliNumber(n)Computes n-th Bernoulli number

Math Constants

eps('mp')Floating-point relative accuracy for current precision
realmax('mp')Largest positive floating-point number with respect to current precision
realmin('mp')Smallest positive floating-point number with respect to current precision
mp('inf') Infinity
mp('nan') Not-a-Number
mp('pi') Ratio of circle’s circumference to its diameter
mp('catalan') Catalan’s constant, K = 0.9159655...
mp('euler') Euler-Mascheroni constant, \gamma = 0.5772156...
mp('ln2') Natural logarithm of 2
i Imaginary unit
j Imaginary unit