# 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.

## 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, $$AX + XB = C$$.

## 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.

## 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. 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.