Mathematical functions

Scientific functions

abs	Returns absolute value of argument.
arccos	Computes inverse cosine.
arcsin	Computes inverse sine.
atan	Computes inverse tangent.
atan2	Computes angle given a point \(x\),:math:y.
besselj	Computes Bessel function, first kind.
besselk	Computes the modified Bessel function of the second kind, \(K_n(x)\).
bessely	Computes Bessel function, second kind.
beta	Computes the complete Beta function, also called the Euler integral.
boxcox	Computes the Box-Cox function.
cos	Computes cosine.
cosh	Computes hyperbolic cosine.
curve	Computes a one-dimensional smoothing curve.
digamma	Computes the digamma function.
exp	Computes the exponential function of \(x\).
fmod	Computes the floating-point remainder of \(x/y\).
gamma	Computes gamma function value.
gammacplx	Computes gamma function for complex inputs.
gammaii	Compute the inverse incomplete gamma function.
ln	Computes the natural log of each element.
lnfact	Computes natural log of factorial function.
lngamma	Computes the natural log of the gamma function.
log	Computes the log (base 10) of each element.
mbesseli	Computes modified and exponentially scaled modified Bessels of the first kind of the nth order.
pi	Returns the constant, \(\pi\).
polygamma	Computes the polygamma function of order \(n\).
psi	Computes the psi (or digamma) function.
sin	Computes sine.
sinh	Computes the hyperbolic sine.
spline	Computes a two-dimensional interpolatory spline.
sqrt	Computes the square root of each element.
tan	Computes tangent.
tanh	Computes hyperbolic tangent.
tocart	Converts from polar to Cartesian coordinates.
topolar	Converts from Cartesian to polar coordinates.
trigamma	Computes trigamma function.
zeta	Computes the Rieman zeta function.

Differentiation and Integration

gradp, gradcplx	Computes first derivative of a function.
hessp, hesscplx	Computes second derivative of a function.
integrate1d	Integrates a user-defined function over a user-defined range, using adaptive quadrature.
intgrat2	Integrates a 2-dimensional function over an user-defined region.
intgrat3	Integrates a 3-dimensional function over an user-defined region.
intquad1	Integrates a 1-dimensional function.
intquad2	Integrates a 2-dimensional function over an user-defined rectangular region.
intquad3	Integrates a 3-dimensional function over an user-defined rectangular region.
intsimp	Integrates by Simpson’s method.

The following are differentiation functions with advanced options.

gradMTm	Computes numerical gradient with mask.
gradMTT	Computes numerical gradient using available threads.
gradMTTm	Computes numerical gradient with mask using available threads.
hessMTg	Computes numerical Hessian using gradient procedure.
hessMTgw	Computes numerical Hessian using gradient procedure with weights.
hessMTm	Computes numerical Hessian with mask.
hessMTmw	Computes numerical Hessian with mask and weights.
hessMTT	Computes numerical Hessian using available threads.
hessMTTg	Computes numerical Hessian using gradient procedure with available threads.
hessMTTgw	Computes numerical Hessian using gradient procedure with weights and using available threads.
hessMTTm	Computes numerical Hessian with mask and available threads.
hessMTw	Computes numerical Hessian with weights.

Linear Algebra

balance	Balances a matrix.
band	Extracts bands from a symmetric banded matrix.
bandchol	Computes the Cholesky decomposition of a positive definite banded matrix.
bandcholsol	Solves the system of equations \(Ax = b\) for \(x\), given the lower triangle of the Cholesky decomposition of a positive definite banded matrix \(A\).
bandltsol	Solves the system of equations \(Ax = b\) for \(x\), where \(A\) is a lower triangular banded matrix
bandrv	Creates a symmetric banded matrix, given its compact form.
bandsolpd	Solves the system of equations \(Ax = b\) for \(x\), where \(A\) is a positive definite banded matrix.
blockDiag	Creates a block-diagonal matrix from one or more input matrices
chol	Computes Cholesky decomposition, \(X=U'U\).
choldn	Performs Cholesky downdate on an upper triangular matrix.
cholsol	Solves a system of equations given the Cholesky factorization of a matrix.
cholup	Performs Cholesky update on an upper triangular matrix.
cond	Computes condition number of a matrix.
crout	Computes Crout decomposition, \(X = LU\) (real matrices only).
croutp	Computes Crout decomposition with row pivoting (real matrices only).
det	Computes determinant of square matrix.
detl	Computes determinant of decomposed matrix.
dot	Returns a scalar dot product of the columns of two matrices.
hess	Computes upper Hessenberg form of a matrix (real matrices only).
inv, invpd	Inverts a square or positive-definite matrices, respectively.
invswp	Computes a generalized sweep inverse.
lapeighb	Computes eigenvalues only of a real symmetric or complex Hermitian matrix selected by bounds.
lapeighi	Computes eigenvalues only of a real symmetric or complex Hermitian matrix selected by index.
lapeighvb	Computes eigenvalues and eigenvectors of a real symmetric or complex Hermitian matrix selected by bounds.
lapeighvi	Computes selected eigenvalues and eigenvectors of a real symmetric or complex Hermitian matrix.
lapgeig	Computes generalized eigenvalues for a pair of real or complex general matrices.
lapgeigh	Computes generalized eigenvalues for a pair of real symmetric or Hermitian matrices.
lapgeighv	Computes generalized eigenvalues and eigenvectors for a pair of real symmetric or Hermitian matrices.
lapgeigv	Computes generalized eigenvalues, left eigenvectors, and right eigenvectors for a pair of real or complex general matrices.
lapgschur	Computes the generalized Schur form of a pair of real or complex general matrices.
lapgsvdcst	Computes the generalized singular value decomposition of a pair of real or complex general matrices.
lapgsvds	Computes the generalized singular value decomposition of a pair of real or complex general matrices.
lapgsvdst	Computes the generalized singular value decomposition of a pair of real or complex general matrices.
ldl	Computes the \(L\) and \(D\) factors of the LDL factorization of a real symmetric matrix.
ldlp	Computes \(LDL\) decomposition with row pivoting of a symmetric matrix.
ldlsol	Computes Solves the system of equations \(LDLTx = b\) using a matrix factorized by ldlp.
lu	Computes \(LU\) decomposition with row pivoting (real and complex matrices).
lusol	Computes Solves the system of equations \(LUx = b\).
norm	Computes one of several specified matrix norms, or a vector p-norm.
null	Computes orthonormal basis for right null space.
null1	Computes orthonormal basis for right null space.
orth	Computes orthonormal basis for column space \(x\).
pinv	Generalized pseudo-inverse: Moore-Penrose.
pinvmt	Generalized pseudo-inverse: Moore-Penrose.
powerM	Computes the power \(n\) of a matrix \(A\), as the mathematical equivalent of the matrix product of \(n\) copies of \(A\).
qqr	\(QR\) decomposition: returns \(Q_1\) and \(R\).
qqre	\(QR\) decomposition: returns \(Q_1\), \(R\) and a permutation vector, \(E\).
qqrep	\(QR\) decomposition with pivot control: returns \(Q_1\), \(R\), and \(E\).
qr	\(QR\) decomposition: returns \(R\).
qre	\(QR\) decomposition: returns \(R\) and \(E\).
qrep	\(QR\) decomposition with pivot control: returns \(R\) and \(E\).
qrsol	Solves a system of equations \(R'x = b\) given an upper triangular matrix, typically the \(R\) matrix from a \(QR\) decomposition.
qrtsol	Solves a system of equations \(Rx = b\) given a lower triangular matrix, typically a transposed \(R\) matrix from a \(QR\) decomposition.
qtyr	\(QR\) decomposition: returns \(Q'Y\) and \(R\).
qtyre	\(QR\) decomposition: returns \(Q'Y\), \(R\) and \(E\).
qtyrep	\(QR\) decomposition with pivot control: returns \(Q'Y\), \(R\) and \(E\).
qyr	\(QR\) decomposition: returns \(QY\) and \(R\).
qyre	\(QR\) decomposition: returns \(QY\), \(R\) and \(E\).
qyrep	\(QR\) decomposition with pivot control: returns \(QY\), \(R\) and \(E\).
qz	Compute the complex \(QZ\), or generalized Schur, form of a pair of real or complex general matrices with an option to sort the eigenvalues.
rank	Computes rank of a matrix.
rref	Computes reduced row echelon form of a matrix.
schtoc	Reduces any 2x2 blocks on the diagonal of the real Schur form of a matrix returned from schur. The transformation matrix is also updated.
schur	Computes real or complex Schur decomposition of a matrix.
solpd	Solves a system of positive definite linear equations.
svd	Computes the singular values of a matrix.
svd1	Computes singular value decomposition, \(X = USV'\).
svd2	Computes the singular value decomposition \(X = USV'\) with compact \(U\).
svdcusv	Computes the singular value decomposition of a matrix so that: \(X = U S V'\) (compact \(U\)).
svds	Computes the singular values of a matrix.
svdusv	Computes the singular value decomposition of a matrix so that: \(X = U S V'\).
sylvester	Computes the solution to the Sylvester matrix equation, \(AX + XB = C\).
tracem	Computes the trace of a matrix.

Eigenvalues

eig	Computes eigenvalues of general matrix.
eigh	Computes eigenvalues of complex Hermitian or real symmetric matrix.
eighv	Computes eigenvalues and eigenvectors of complex Hermitian or real symmetric matrix.
eigv	Computes eigenvalues and eigenvectors of general matrix.

Polynomial Operations

polychar	Computes characteristic polynomial of a square matrix.
polyeval	Evaluates polynomial with given coefficients.
polyint	Calculates Nth order polynomial interpolation given known point pairs.
polymake	Computes polynomial coefficients from roots.
polymat	Returns sequence powers of a matrix.
polymult	Multiplies two polynomials together.
polyroot	Computes roots of polynomial from coefficients.

See also recserrc, recsercp, and conv.

Fourier Transforms

dfft	Computes discrete 1-D FFT.
dffti	Computes inverse discrete 1-D FFT.
fft	Computes 1- or 2-D FFT.
ffti	Computes inverse 1- or 2-D FFT.
fftm	Computes multi-dimensional FFT.
fftmi	Computes inverse multi-dimensional FFT.
fftn	Computes 1- or 2-D FFT using prime factor algorithm.
rfft	Computes real 1- or 2-D FFT.
rffti	Computes inverse real 1- or 2-D FFT.
rfftip	Computes inverse real 1- or 2-D FFT from packed format FFT.
rfftn	Computes real 1- or 2-D FFT using prime factor algorithm.
rfftnp	Computes real 1- or 2-D FFT using prime factor algorithm, returns packed format FFT.
rfftp	Computes real 1- or 2-D FFT, returns packed format FFT.

Fuzzy Conditional Functions

dotfeq, dotfge, dotfgt, dotfle, dotflt, dotfne	Fuzzy .==, .>=, .>, .<=, .<, .!=
dotfeqmt, dotfgemt, dotfgtmt, dotflemt, dotfltmt, dotfnemt	Fuzzy .==, .>=, .>, .<=, .<, .!=
feq, fge, fgt, fle, flt, fne	Fuzzy ==, >=, >, <=, <, !=
feqmt, fgemt, fgtmt, flemt, fltmt, fnemt	Fuzzy ==, >=, >, <=, <, !=

The mt commands use an fcmptol argument to control the tolerance used for comparison. The non-mt commands use the global variable _fcmptol to control the tolerance used for comparison. By default, this is 1e-15. The default can be changed by editing the file fcompare.dec.

Statistical Functions

acf	Computes sample autocorrelations.
astd	Computes the standard deviation of the elements across one dimension of an N-dimensional array.
astds	Computes the ‘sample’ standard deviation of the elements across one dimension of an N-dimensional array.
chiBarSquare	Computes probability of chi-bar-square statistic.
clusterSE	Computes the White cluster-robust standard errors.
combinate	Computes combinations of \(n\) things taken \(k\) at a time.
combinated	Writes combinations of \(n\) things taken \(k\) at a time to a GAUSS data set.
ConScore	Computes constrained score statistic and its probability.
conv	Computes convolution of two vectors.
corrm, corrvc, corrx	Computes an unbiased estimate of a correlation matrix from a moment matrix, variance-covariance matrix or general matrix.
crossprd	Computes cross product.
design	Creates a design matrix of 0’s and 1’s.
dstatmt	Computes descriptive statistics of a data set or matrix.
dwstat	Computes the Durbin-Watson statistic.
dot	Computes a scalar dot product of the columns of two matrices.
gdaDStat	Computes descriptive statistics on multiple Nx1 variables in a GDA.
gdaDStatMat	Computes descriptive statistics on a selection of columns in a variable in a GDA.
glm	Computes generalized linear regression of a matrix.
gmmFit	Computes generalized method of moments estimates from user specified moment function.
gmmFitIV	Estimate instrumental variables model using the generalized method of moments.
loess	Computes coefficients of locally weighted regression.
loessmt	Computes coefficients of locally weighted regression.
meanc	Computes mean value of each column of a matrix.
median	Computes medians of the columns of a matrix.
moment	Computes moment matrix (\(x'x\)) with special handling of missing values.
momentd	Computes moment matrix from a data set.
movingave	Computes moving average of a series.
movingaveExpwgt	Computes exponentially weighted moving average of a series.
movingaveWgt	Computes weighted moving average of a series.
numCombinations	Computes number of combinations of \(n\) things taken \(k\) at a time.
ols	Computes least squares regression of data set or matrix.
olsmt	Computes least squares regression of data set or matrix.
olsqr	Computes OLS coefficients using \(QR\) decomposition.
olsqr2	Computes OLS coefficients, residuals, and predicted values using \(QR\) decomposition.
olsqrmt	Computes OLS coefficients using \(QR\) decomposition.
pacf	Computes sample partial autocorrelations.
princomp	Computes principal components of a data matrix.
quantile	Computes quantiles from data in a matrix, given specified probabilities.
quantiled	Computes quantiles from data in a data set, given specified probabilities.
quantileFit	Perform linear quantile regression.
quantileFitLoc	Perform local linear or quadratic quantile regression.
robustSE	Computes the Huber-White heteroscedastic robust standard errors. The procedure uses the “sandwich” variance-covariance estimator with a small sample correction of \((n)/(n-1)\).
stdc	Computes standard deviation of the columns of a matrix.
toeplitz	Computes Toeplitz matrix from column vector.
varmall	Computes the log-likelihood of a Vector ARMA model.
varmares	Computes the residuals of a Vector ARMA model.
vcm, vcx	Computes an unbiased estimate of a variance-covariance matrix from a matrix \(x\) or a moment matrix, \(x'x\).

Series and Sequence Functions

recserar	Computes autoregressive recursive series.
recsercp	Computes recursive series involving products.
recserrc	Computes recursive series involving division.
recserVAR	Computes a vector autoregressive recursive.
seqa, seqm	Creates an additive or multiplicative sequence, respectively.
seqaDT	Creates a sequence of dates in DT scalar format.
seqaPosix	Creates a sequence of dates in posix date format, returned as a dataframe date variable.

Precision Control

base10	Converts number to x.xxx and a power of 10.
ceil	Rounds up towards \(+\infty\).
fix	Rounds towards 0.
floor	Rounds down towards \(-\infty\).
machEpsilon	Returns the smallest number such that \(1 + eps > 1\).
round	Rounds to the nearest integer.
trunc	Converts numbers to integers by truncating the fractional portion.