Q Math Library


qml is a library for statistics, linear algebra, and optimization in kdb+. It provides an interface between the q programming language and numerical libraries such as LAPACK.

The current version of the library is 0.5.2 (track updates). This is a development branch. For precompiled binaries see version 0.2.1.



The latest qml 0.5 wraps up some work I started over three years ago, right after releasing qml 0.2.1. The idea is for the optimization routines .qml.min and .qml.conmin to become a single uniform interface to different underlying libraries such as CONMAX and NLopt.

NLopt provides qml with derivative-free algorithms for unconstrained and constrained optimization. While CONMAX, the only algorithm available previously, also works without explicitly specified gradient functions, it has to approximate them with central differences.


Welcome qml 0.4. Development continues on GitHub.


I am working on a new version of QML.

The interface will be the same as in version 0.3, including the recently added mm, ms, mls and mlsq functions, while the configuration will be more like version 0.2, with BLAS used by default and precompiled binaries provided.

It will be compatible with kdb+ 3.0, testing will be more thorough than ever, and there will be an option to link against a different implementation of BLAS, such as ATLAS.

In the mean time, check out this patch from Kim Tang for kdb+ 3.0 support.

Also, if you want to build QML 0.3.10, the particular development snapshot of ATLAS that it uses is no longer available for download, so I'm hosting it here.

archive atlas-3.9.46.tar.bz2 (5.31 MB)

Just verified that it still builds on 32-bit Debian.


QML 0.2.1 is still the stable version. It does not have the matrix performance of ATLAS, but it is easier to compile and includes binaries for 32-bit platforms.

Since it was released some of the files on the Netlib site changed, and QML 0.2.1 no longer compiles with them. I'm hosting the original files here. Place them in the download subdirectory before running make.

archive clapack.tgz (6.89 MB)

archive f2c.tar.gz (240.00 KB)

Earlier entries


qml is free software with a BSD-style license. It is provided in the hope that it will be useful, but with absolutely no warranty. See the file LICENSE.txt for details.

qml is linked against several other libraries. The copyrights and licenses for these libraries are also listed in LICENSE.txt.


Current version:

repository Source code on GitHub

archive qml-0.5.2.zip (160.89 KB; 2014-06-16; source code only)

Older version:

archive qml-0.4.zip (99.65 KB; 2013-05-12; source code only)

Legacy versions:

repository Source code on GitHub

archive qml-0.3.10.zip (76.88 KB; 2011-07-13; source code only)

archive qml-0.3.8.zip (74.28 KB; 2011-07-08; source code only)

archive qml-0.2.1.zip (1.31 MB; 2010-02-17; includes binaries)

archive qml-0.1.8.zip (1.23 MB; 2009-10-25; includes binaries)


To compile and install from source code, run

make test
make install

To install a precompiled binary, copy qml.q into the same directory as q.k, and copy qml.dll or qml.so into the same directory as q.exe or q. Then run test.q.


Load with

q)\l qml.q

All functions are in the .qml namespace. Numerical arguments are automatically converted into floating-point. Matrixes are in the usual row-major layout (lists of row vectors). Complex numbers are represented as pairs of their real and imaginary parts.

q).qml.nicdf .25 .5 .975                  / normal distribution quantiles
-0.6744898 0 1.959964

q).qml.mchol (1 2 1;2 5 4;1 4 6)          / Cholesky decomposition
1 2 1
0 1 2
0 0 1

q).qml.poly 2 -9 16 -15                   / solve 2x^3-9x^2+16x-15=0
1 1.414214
1 -1.414214

q).qml.mlsq[(1 1;1 2;1 3;1 4);11 2 -3 -4] / fit line
14 -5f

q).qml.conmin[{x*y+1};{1-(x*x)+y*y};0 0]  / minimize x(y+1) s.t. x^2+y^2<=1
-0.8660254 0.5

There are more examples here and in my directory at code.kx.com.

Constants and functions


pi pi
e e
eps smallest representable step from 1.

Trigonometric functions

sin[x] sine
cos[x] cosine
tan[x] tangent
asin[x] arcsine
acos[x] arccosine
atan[x] arctangent
atan2[x;y] atan[x%y]
sinh[x] hyperbolic sine
cosh[x] hyperbolic cosine
tanh[x] hyperbolic tangent
asinh[x] hyperbolic arcsine
acosh[x] hyperbolic arccosine
atanh[x] hyperbolic arctangent

Other libm functions

exp[x] exponential
expm1[x] exp[x]-1
log[x] logarithm
log10[x] base-10 logarithm
logb[x] extract binary exponent
log1p[x] log[1+x]
pow[a;x] exponentiation
sqrt[x] square root
cbrt[x] cube root
hypot[x;y] sqrt[pow[x;2]+pow[y;2]]
floor[x] round downward
ceil[x] round upward
fabs[x] absolute value
fmod[x;y] remainder of x%y

Hypergeometric functions

erf[x] error function
erfc[x] complementary error function
lgamma[x] log of absolute value of gamma function
gamma[x] gamma function
beta[x;y] beta function
pgamma[a;x] lower incomplete gamma function (a>0)
pgammac[a;x] upper incomplete gamma function (a>0)
pgammar[a;x] regularized lower incomplete gamma function (a>0)
pgammarc[a;x] regularized upper incomplete gamma function (a>0)
ipgammarc[a;p] inverse complementary regularized incomplete gamma function (a>0, p0.5)
pbeta[a;b;x] incomplete beta function (a,b>0)
pbetar[a;b;x] regularized incomplete beta function (a,b>0)
ipbetar[a;b;p] inverse regularized incomplete beta function (a,b>0)
j0[x] order 0 Bessel function
j1[x] order 1 Bessel function
y0[x] order 0 Bessel function of the second kind
y1[x] order 1 Bessel function of the second kind

Probability distributions

ncdf[x] CDF of normal distribution
nicdf[p] its inverse
c2cdf[k;x] CDF of chi-squared distribution (k1)
c2icdf[k;p] its inverse
stcdf[k;x] CDF of Student's t-distribution (natural k)
sticdf[k;p] its inverse
fcdf[d1;d2;x] CDF of F-distribution (d1,d21, x0)
ficdf[d1;d2;p] its inverse
gcdf[k;th;x] CDF of gamma distribution
gicdf[k;th;x] its inverse
bncdf[k;n;p] CDF of binomial distribution
bnicdf[k;n;x] its inverse for p parameter (k<n)
pscdf[k;lambda] CDF of Poisson distribution
psicdf[k;p] its inverse for lambda
smcdf[n;e] CDF for one-sided Kolmogorov-Smirnov test
smicdf[n;e;x] its inverse
kcdf[x] CDF for Kolmogorov distribution
kicdf[p] its inverse (p1e-8)

Matrix operations

diag[diag] make diagonal matrix
mdim[matrix] number of (rows; columns)
mdiag[matrix] extract main diagonal
mdet[matrix] determinant
mrank[matrix] rank
minv[matrix] inverse
mpinv[matrix] pseudoinverse
mm[A;B] multiply
ms[A;B] solve B=A mm X, A is triangular
mev[matrix] (eigenvalues; eigenvectors) sorted by decreasing modulus
mchol[matrix] Cholesky factorization upper matrix
mqr[matrix] QR factorization: (Q; R)
mqrp[matrix] QR factorization with column pivoting: (Q; R; P), matrix@\:P=Q mm R
mlup[matrix] LUP factorization with row pivoting: (L; U; P), matrix[P]=L mm U
msvd[matrix] singular value decomposition: (U; Sigma; V)
mkron[A;B] Kronecker product

Polynomial roots

poly[coef] roots of a polynomial (highest-degree coefficient first, can be complex)

Linear equation solving

mls[A;B] solve B=A mm X
mlsx[opt;A;B] mls[] with options
`equi: equilibrate the system (default: don't)
mlsq[A;B] solve min ||B-A mm X||
mlsqx[opt;A;B] mlsq[] with options
`svd: use SVD algorithm (default: QR or LQ)

Nonlinear equation solving and optimization

root[f;(x0;x1)] find root on interval (f(x0)f(x1)<1)
rootx[opt;f;(x0;x1)] root[] with options (as dictionary or mixed list)
`iter: max iterations (default: 100)
`tol: numerical tolerance (default: ~1e-8)
`full: full output (default: only x)
`quiet: return null on failure (default: signal)
solve[eqs;x0] solve nonlinear equations (given as functions)
solvex[opt;eqs;x0] solve[] with options
`iter: max iterations (default: 1000)
`tol: numerical tolerance (default: ~1e-8)
`full: full output (default: only x)
`quiet: return null on failure (default: signal)
`steps: RK steps per iteration (default: 1)
`rk: use RK steps only (default: RK, SLP)
`slp: use SLP steps only (default: RK, SLP)
line[f;base;x0] line search for minimum from base
linex[opt;f;base;x0] line[] with same options as rootx[]
min[f;x0] find unconstrained minimum
minx[opt;f;x0] min[] with same options as solvex[], plus
`nm: use Nelder-Mead method (default: CONMAX)
`sbplx: use Subplex method (default: CONMAX)
conmin[f;cons;x0] find constrained minimum (functions cons0)
conminx[opt;f;cons;x0] min[] with same options as solvex[], plus
`lincon: assume linear cons (default: nonlinear)
`cobyla: use COBYLA method (default: CONMAX)