# LACAML - Linear Algebra for OCaml

## What is LACAML?

This OCaml-library interfaces two widely used mathematical FORTRAN-libraries:

This allows developers to write high-performance numerical code for applications that require linear algebra.

## Features

• The BLAS- and LAPACK-libraries have evolved over about two decades of time and are therefore extremely mature both in terms of stability and performance.

• LACAML interfaces most of the functions in BLAS and LAPACK (many hundreds!). It supports among other things linear equations, least squares problems, eigenvalue problems, singular value decomposition (SVD), Cholesky and QR-factorization, etc.

• Many convenience functions for creating and manipulating matrices.

• Powerful printing functions for large vectors and matrices and supplemental information (e.g. row and column headers). Users can specify easily how much context to print. For example, it is usually sufficient to print small blocks of the four corners of a large result matrix to manually verify the correctness of an algorithm. LACAML uses this approach to limit the output to human-manageable size.

• Integration into the OCaml-toplevel allows for easy experimentation for students and researchers as well as demonstration for lecturers. Values of vector and matrix type will be printed automatically without cluttering the screen.

• The OCaml-interface was designed in a way to combine both the possibility of gaining optimum efficiency (e.g. by allowing the creation of work arrays outside of loops) with simplicity (thanks to labels and default arguments).

• The code is precision-independent and supports both real and complex transforms in a consistent way. There are four modules that implement the same interface modulo the precision type and specialized real/complex functions. If you refer to elements in this interface only, your code becomes precision- and (if meaningful) real/complex independent, too: you can choose at anytime whether you want to use single-precision or double-precision simply by referring to the required module.

• You can fully exploit the library within multithreaded programs. Many numerical routines are likely to run for a long time, but they will never block other threads. This also means that you can execute several routines at the same time on several processors if you use POSIX-threads in OCaml.

• To make things easy for developers used to the "real" implementation in FORTRAN but also for beginners who need detailed documentation, both function- and argument names have been kept compatible to the ones used in the BLAS- and LAPACK-documentation. Only exception: you need not prefix functions with `s`, `d`, `c` or `z` to indicate the precision and type of numbers, because the OCaml module system provides us with a more convenient means of choosing them.

• (Almost) all errors are handled within OCaml. Typical mistakes like passing non-conforming matrices, parameters that are out of range, etc., will be caught before calling Fortran code and will raise exceptions. These exceptions will explain the error in detail, for example the received illegal parameter and the range of expected legal values.

The only exception to the above is illegal contents of vectors and matrices. This can happen, for example, when freshly allocated matrices are used without initialization. Some LAPACK-algorithms may not be able to deal with floats that correspond to NaNs, infinities, or are subnormal. Checking matrices on every call would seem excessive. Some functions also expect matrices with certain properties, e.g. positive-definiteness, which would be way too costly to verify beforehand. It is the task of the user to make sure that data contained in matrices is suitable for the application of the intended functions.

## Using LACAML

You can make use of this library by referring to the corresponding module for the required precision and number type. E.g.:

```open Lacaml.S  (* Single-precision real numbers *)
open Lacaml.D  (* Double-precision real numbers *)
open Lacaml.C  (* Single-precision complex numbers *)
open Lacaml.Z  (* Double-precision complex numbers *)
```

These modules become available if you link the `lacaml`-library with your application. The widely used OCaml-tool `findlib` will take care of linking `lacaml` correctly. If you do not use this tool, you will also have to link in the `bigarray`-library provided by the OCaml-distribution.

The `Lacaml.?`-modules implement the BLAS/LAPACK-interface. Their corresponding submodules `Vec` and `Mat` provide for vector and matrix operations that relate to the given precision and number type.

Most functions were implemented using optional arguments (= default arguments). If you do not provide them at the call-site, sane defaults will be used instead. Here is an example of a function call:

```let rank = gelss in_mat out_mat in
(* ... *)
```

This example computes the solution to a general least squares problem (= linear regression) using the SVD-algorithm with `in_mat` as the matrix containing the predictor variables and `out_mat` as the matrix containing (possibly many) response variables (this function can handle several response variables at once). The result is the rank of the matrix. The matrices provided in the arguments will be overwritten with further results (here: the singular vectors and the solution matrix).

If the above happened in a loop, this would be slightly inefficient, because a work-array would have to be allocated (and later deallocated) at each call. You can hoist the creation of this work array out of the loop, e.g. (`m`, `n`, `nrhs` are problem dependent parameters):

```let work = gelss_min_work ~m ~n ~nrhs in
for i = 1 to 1000 do
(* ... *)
let rank = gelss in_mat ~work out_mat in
(* ... *)
done
```

All matrices can be accessed in a restricted way, i.e. you can specify submatrices for all matrix parameters. For example, if some matrix is called `a` in the interface documentation, you can specify the left upper corner of the wanted submatrix for the operation by setting `ar` for the row and `ac` for the column (1 by default). A vector `y` would have an extra optional parameter `ofsy` (also 1 by default). Parameters like `m` or `n` typically specify the numbers of rows or columns.

### Printing vectors and matrices

Here is a toplevel example of printing a large random matrix:

```# #require "lacaml";;
# open Lacaml.D;;
# let mat = Mat.random 100 200;;
val mat : Lacaml.D.mat =
C1        C2        C3          C198       C199      C200
R1 -0.314362 -0.530711  0.309887 ...  0.519965  -0.230156 0.0479154
R2  0.835658  0.581404  0.161607 ... -0.749358  -0.630019 -0.858998
R3 -0.403421  0.458116 -0.497516 ...  0.210811   0.422094  0.589661
...       ...       ... ...       ...        ...       ...
R98 -0.352474  0.878897  0.357842 ...  0.150786   -0.74011  0.353253
R99  0.104805  0.984924 -0.319127 ... -0.143679  -0.858269  0.859059
R100  0.419968  0.333358  0.237761 ... -0.483535 -0.0224016  0.513944
```

Only the corner sections of the matrix, which would otherwise be too large to display readably, are being printed, and ellipses (`...`) are used in place of the removed parts of the matrix.

If the user required even less context, the `Lacaml.Io.Toplevel.lsc` function, which is also available in each precision module for convenience (here: `Lacaml.D`), could be used to indicate how much. In the following example only two-by-two blocks are requested in each corner of the matrix:

```# lsc 2;;
- : unit = ()
# mat;;
- : Lacaml.D.mat =
C1        C2           C199      C200
R1 -0.314362 -0.530711 ...  -0.230156 0.0479154
R2  0.835658  0.581404 ...  -0.630019 -0.858998
...       ... ...        ...       ...
R99  0.104805  0.984924 ...  -0.858269  0.859059
R100  0.419968  0.333358 ... -0.0224016  0.513944
```

Applications can use the standard `Format`-module in the OCaml-distribution together with LACAML printing functions to output vectors and matrices. Here is an example using labels and showing the high customizability of the printing functions:

```open Lacaml.D
open Lacaml.Io

let () =
let rows, cols = 200, 100 in
let a = Mat.random rows cols in
Format.printf "@[<2>This is an indented random matrix:@\n@\n%a@]@."
(Lacaml.Io.pp_lfmat
~row_labels:
(Array.init rows (fun i -> Printf.sprintf "Row %d" (i + 1)))
~col_labels:
(Array.init cols (fun i -> Printf.sprintf "Col %d" (i + 1)))
~vertical_context:(Some (Context.create 2))
~horizontal_context:(Some (Context.create 3))
~ellipsis:"*"
~print_right:false
~print_foot:false ())
a
```

The above code might print:

```This is an indented random matrix:

Col 1     Col 2       Col 3      Col 98    Col 99   Col 100
Row 1  0.852078 -0.316723    0.195646 *  0.513697  0.656419  0.545189
Row 2 -0.606197  0.411059    0.158064 * -0.368989    0.2174    0.9001
*         *           * *         *         *         *
Row 199 -0.684374 -0.939027 0.000699582 *  0.117598 -0.285587 -0.654935
Row 200  0.929341 -0.823264    0.895798 *  0.198334  0.725029 -0.621723
```

Many other options, e.g. for different padding, printing numbers in other formats or with different precision, etc., are available for output customization.

### Other sources of usage information

#### API documentation

Please refer to the API-documentation that will be generated as HTML-files if requested (`make doc`). It will give more details on the numerous functions and supported arguments in the library. It can also be found online.

#### BLAS/LAPACK man pages

BLAS and LAPACK binary packages for Unix operating systems usually come with appropriate man-pages. E.g. to quickly look up how to factorize a positive-definite, complex, single precision matrix, you might enter:

```man cpotrf
```

The corresponding function in Lacaml would be `Lacaml.C.potrf`. The naming conventions and additional documentation for BLAS and LAPACK can be found at their respective websites.

#### Examples

The `examples`-directory contains several demonstrations of how to use this library for various linear algebra problems.

## Improving Performance

It is highly recommended that users install a variant of BLAS (or even LAPACK) that has been optimized for their system. Processor vendors (e.g. Intel) usually sell the most optimized implementations for their CPU-architectures. Some computer and OS-vendors like Apple distribute their own implementations with their products (e.g. `vecLib`).

There is also ATLAS, a very efficient and compatible substitute for BLAS. It specializes code for the architecture it is compiled on. Binary packages (e.g. RPMs) for Linux should be available from your distribution vendor's site (you must recompile the package to make sure it is suited to your distribution, see the package documentation for more details.).

Another alternative for BLAS is OpenBLAS.

## Contact Information and Contributing

Up-to-date information concerning this tool should be available at: https://bitbucket.org/mmottl/lacaml

Enjoy!

Markus Mottl in Rutherford, NJ on July 9, 2012

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