Q Math Library
Introduction
The Q Math Library provides the q programming language and KDB+ database with an interface to a number of useful mathematical functions from the libm, Cephes, ATLAS, LAPACK and CONMAX libraries. The source code, and especially the build system, may also be of interest to other developers of libraries for KDB+.
The current version of the library is 0.3.10 (track updates). This is a development branch and tested only on 32-bit Linux. For precompiled binaries see version 0.2.1.
News
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License
The Q Math Library is free software, available under a BSD-style license. It is provided in the hope that it will be useful, but without any warranty; without even the implied warranties of merchantability and fitness for a particular purpose. See the file LICENSE.txt in the distribution for more information.
This library is intended to be linked, and the precompiled binaries are linked, against several other libraries. The copyrights and licenses for these libraries are also listed in the file LICENSE.txt.
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Usage
Two files make up the library: qml.dll (in a platform-specific subdirectory) and qml.q. Put them both somewhere where q can find them and load with
q)\l qml.q
All functions will be in the .qml namespace. The functions accept any numerical arguments and convert them into floating-point. Matrixes are in row-major order, as usual. Complex numbers are represented as pairs of the real and imaginary parts. E.g.:
q).qml.nicdf .25 .5 .75 / normal distribution quartiles -0.6744898 0 0.6744898 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 2.5 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.
It's recommended to run the test suite, test.q, to make sure that everything is working correctly.
Building the library requires GNU make and GCC. On Windows, these are provided by Cygwin and MinGW (or MinGW-w64). The build, just run make. See the top of the Makefile for options.
make make test
The included binaries are currently compiled with GCC 4.4 from MinGW-w64 on Windows, GCC 4.4 on Debian Linux and GCC 3.3 on Darwin.
Constants and functions
Constants
| 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, p≥0.5) |
| 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] | inverse CDF of normal distribution |
| nicdf[p] | inverse CDF of normal distribution |
| c2cdf[k;x] | inverse CDF of chi-squared distribution (k≥1) |
| c2icdf[k;p] | inverse CDF of chi-squared distribution (k≥1) |
| stcdf[k;x] | inverse CDF of Student's t-distribution (natural k) |
| sticdf[k;p] | inverse CDF of Student's t-distribution (natural k) |
| fcdf[d1;d2;x] | inverse CDF of F-distribution (d1,d2≥1, x≥0) |
| ficdf[d1;d2;p] | inverse CDF of F-distribution (d1,d2≥1, x≥0) |
| gcdf[k;th;x] | inverse CDF of gamma distribution |
| gicdf[k;th;x] | inverse CDF of gamma distribution |
| bncdf[k;n;p] | inverse CDF of binomial distribution |
| bnicdf[k;n;x] | inverse CDF of binomial distribution for p parameter (k<n) |
| pscdf[k;lambda] | inverse CDF of Poisson distribution |
| psicdf[k;p] | inverse CDF of Poisson distribution for lambda parameter |
| smcdf[n;e] | inverse CDF for one-sided Kolmogorov-Smirnov test |
| smicdf[n;e;x] | inverse CDF for one-sided Kolmogorov-Smirnov test |
| kcdf[x] | inverse CDF for Kolmogorov distribution |
| kicdf[p] | inverse CDF for Kolmogorov distribution (p≥1e-8) |
Matrix operations
| diag[diag] | make diagonal matrix |
| 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) |
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
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| mlsq[A;B] | solve min ||B=A mm X|| | |||
| mlsqx[opt;A;B] | mlsq[] with options
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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)
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| solve[eqs;x0] | solve nonlinear equations (given as functions) | |||||||||||||||||||||
| solvex[opt;eqs;x0] | solve[] with options
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| 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[] | |||||||||||||||||||||
| conmin[f;cons;x0] | find constrained minimum (functions cons≥0) | |||||||||||||||||||||
| conminx[opt;f;cons;x0] | min[] with same options as solvex[], plus
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