# Fast, Native Speed, Vector Computations in Clojure

## Vector and Matrix Data Structures

Before we do any numeric computations, we have to create the data to run the computations on.

Neanderthal supports any pluggable infrastructure (GPU computation is already available!), and the default is to use vectors and matrices backed by direct byte buffers, that can be sent to native libraries or GPU via JNI without copying overhead.

### Creating Vectors and Matrices

Functions for creating the appropriate primitive vectors or matrices are
in the `uncomplicate.neanderthal.native`

namespace. Additional implementations
are available in appropriate namespaces; for example,
`uncomplicate.neanderthal.opencl`

activates a GPU accelerated
implementation.

Import the appropriate namespaces: `core`

for computation functions,
and `native`

for constructors.

```
(ns uncomplicate.neanderthal.examples.guides.tutorial-native-test
(:require [midje.sweet :refer :all]
[uncomplicate.fluokitten.core :refer [fmap! fold]]
[uncomplicate.neanderthal
[block :refer [buffer]]
[core :refer :all]
[native :refer :all]])
(:import [java.nio ByteBuffer ByteOrder]
clojure.lang.ExceptionInfo))
```

Creation functions follow the BLAS naming scheme: d for doubles, f for floats (instead of BLAS’s s), c for complex, ge for general dense matrix etc:

`dv`

creates a vector of doubles`fv`

creates a vector of floats`dge`

creates a matrix of doubles etc.

This tutorial will work with double-precision floats. Single-precision floats are used in exactly the same way, except for the constructors. Single precision does not matter much on the CPU (other that the performance of BLAS computations may be up to 2x faster and they use 2x less storage space) but on the GPU it is the preferred format, since current consumer-grade GPUs usually offer pro-grade performance in single precision while being crippled for double precision 8x or, recently, 32x.

All numbers in Neanderthal, both the data it holds and numbers that
the functions return are **primitive** where it matters (more about that later).

Here are a few examples written as Midje tests:

```
(facts
"We create a few double vectors using different input methods."
(dv 10) => (dv (repeat 10 0))
(dv 10.0) => (dv [10])
(dv '(1 2 3 4 5 6)) => (dv 1 2 3 4 5 6)
(dv (dv (repeat 10 0))) => (dv (repeat 10 0)))
(facts
"And here is how you create double general dense matrices."
(dge 2 3) => (dge 2 3 (repeat 6 0))
(dge 3 2 [1 2 3 4 5 6]) => (dge 3 2 '(1 2 3 4 5 6))
(dge 2 3 (dge 2 3 (repeat 6 0))) => (dge 2 3 (repeat 6 0)))
```

### Neanderthal keeps data in direct byte buffers

Please note that a non-buffer input source (numbers, varargs, sequences) is suitable only as a convenience for smallish data and test code. Be careful about the performance when working with large data, though

- sequences are slow and contain boxed numbers! Thus, the preferred way
for fast population of your matrices is to create a raw matrix, and use entry!,
or to extract its raw ByteBuffer, and work on it
**carefully**.

It is awkward and cumbersome to directly work with buffers. You should take care of endianess: java uses BIG_ENDIAN, while Intel processors and most native platforms natively support LITTLE_ENDIAN. If you pre-load your data in buffers, you, or the library you use, have to take care of using the proper native endianess. Also take care to revert the buffer position to 0. Neanderthal does not care how you prepare the buffers as long as the data is prepared well. You can use some of the existing libraries that work with native buffers (Vertigo, etc.), check out Neanderthal API to see what utilities are currently available, or roll your own.

Matrix data is also kept in a one-dimensional byte buffer, and NOT in a object buffer or array that holds raw buffers, for performance reasons. By default, when used in 2D matrices, Neanderthal treats a 1D buffer as a sequence of columns. Column-oriented order is commonly used in numerical software, contrary to row-oriented order used by the C language. Java uses neither; 2D arrays are arrays of array references, and this difference has a huge performance impact. Neanderthal abstracts all these performance optimizations away, and you do not need to care about this, unless you write a pluggable Neanderthal implementation. When you need to harness other structures, Neanderthal’s constructors take additional options, though!

The same ByteBuffer can hold data for vectors as well as matrices.

```
(facts
"Here is what you need to take care of if you opt to provide the initial data
in raw byte buffers."
(let [entry-width Double/BYTES
m 2
n 3
empty-matrix (dge m n)
empty-buf (buffer empty-matrix)
filled-buf (loop [i 0 buf empty-buf]
(if (< i (* m n))
(recur (inc i) (.putDouble ^ByteBuffer buf (double i)))
buf))
rewind-buf (.position ^ByteBuffer filled-buf 0)]
empty-matrix => (dge 2 3 (range (* m n)))))
```

## Pure and Non-pure Functions

Many BLAS functions work in-place! It means that they mutate the data they work
on. That way, they are orders of magnitude faster and use less memory. These
functions have a BANG (`!`

) suffix. They have non-destructive, pure variants,
without the `!`

. Keep in mind that both variants are useful for specific tasks.
Usually, we should use the destructive variants for fast algorithm internals or
with huge data when there is no space for copying.

## BLAS Level 1 Functions

BLAS Level 1 contains functions that compute in linear time, O(n). They usually work with 1D vectors, but some of them are also appropriate for 2D matrices. Some of these functions compute a number based on vector entries while some transform the values of entries.

I will show you the most popular ones, so you can easily find your way with others in Neanderthal API docs.

```
(facts
"BLAS 1 asum: Sum of absolute values."
(asum (dv 1 2 -5)) => 8.0)
(facts
"BLAS 1 sum: Sum of all values. Note: this function is not a part of
BLAS standard."
(sum (dv 1 2 -5)) => -2.0)
(facts
"BLAS 1 dot: Dot product is a sum of the scalar products of respective entries
of two vectors."
(dot (dv 1 2 3) (dv 1 3 5)) => 22.0)
(facts
"BLAS 1 copy: Here is how we copy the data from one vector to another. We may
provide the destionation and change it, or copy the data into a new vector.
And, it works with matrices, too."
(let [x (dv 1 2 3)
y (dv 3)
a (dge 2 3 (range 6))
b (dge 2 3)]
(copy! x y) => (dv 1 2 3)
y => (dv 1 2 3)
(copy x) => (dv 1 2 3)
(copy! a b) => (dge 2 3 (range 6))
b => (dge 2 3 (range 6))))
(facts
"BLAS 1 swap: Here is how we destructively swap the entries of two vectors or
matrices."
(let [x (dv 1 2 3)
y (dv 1 3 5)
a (dge 2 3 (range 6))
b (dge 2 3)]
(swp! x y) => x
x => (dv 1 3 5)
y => (dv 1 2 3)
(swp! a b) => a
a => (dge 2 3)
b => (dge 2 3 (range 6))))
(facts
"BLAS 1 scal: Here is how you scale a vector (multiply all entries in a vector
by a scalar value). Also works on matrices."
(let [x (dv 1 2 3 4 5)]
(scal! 3.0 x) => (dv 3 6 9 12 15)
x => (dv 3 6 9 12 15)))
(facts
"BLAS 1 axpy: SAXPY stands for Scalar a times x plus y. It scales a vector and
adds it to another vector. It can help in acomplishing both the scaling of
vectors, additions, or both. It also works on matrices.
It have destructive and non-destructive variants, and accepts varargs:
ax - scaling
xpy - vector addition
axpy - scaling and addition
axpy! - destructive scaling and addition
"
(let [x (dv 1 2 3)
y (dv 1 3 5)]
(ax 2.5 x) => (dv 2.5 5.0 7.5)
x => (dv 1 2 3)
(xpy x y) => (dv 2 5 8)
(xpy x y (dv 3 2 1) x) => (dv 6 9 12)
x => (dv 1 2 3)
(axpy x y) => (xpy x y)
(axpy x y (dv 3 2 1) x) => (xpy x y (dv 3 2 1) x)
(axpy 1.5 x (zero x) 2.5 y) => (dv 4 10.5 17)
(axpy 2.5 y x (dv 1 2 2) 1.5 (dv 2 1 3)) => (dv 7.5 13 22)
(axpy! 1.5 x y) => (dv 2.5 6 9.5)
y => (dv 2.5 6 9.5)
(axpy! 2.5 x y (dv 1 2 1) 1.4 (dv 2 3 4)) => (dv 8.8 17.2 23.6)))
```

## BLAS Level 2 Functions

BLAS Level 2 functions are those that compute in quadratic time O(n^2). Usually, these functions combine matrices and vectors.

```
(facts
"BLAS 2 mv: Here is how to perform a matrix-vector multiplication.
mv! is a destructive version, while mv always returns the result in a new vector
instance."
(let [m (dge 2 3 (range 6))
v (dv 1 2 3)
y (dv 1 2)]
(mv m v) => (dv 16 22)
(mv m (dv 1 2)) => (throws ExceptionInfo)
(mv 1.5 m v) => (dv 24 33)
m => (dge 2 3 (range 6))
v => (dv 1 2 3)
y => (dv 1 2)
(mv! 1.5 m v 2.5 y) => (dv 26.5 38.0)
y => (dv 26.5 38.0)))
(facts
"BLAS 2 rk: Here is how to multiply a transposed vector by another vector,
thus creating a matrix. rk! puts the result in a provided matrix, while rk
puts it in a new matrix instance."
(let [a (dge 2 3 (range 6))
x (dv 1 2)
y (dv 1 2 3)]
(rk x y) => (dge 2 3 [1 2 2 4 3 6])
(rk 1.5 x y) => (dge 2 3 [1.5 3.0 3.0 6.0 4.5 9.0])
a => (dge 2 3 (range 6))
(rk! 2.5 x y a) => (dge 2 3 [2.5 6.0 7.0 13.0 11.5 20.0])
a => (dge 2 3 [2.5 6.0 7.0 13.0 11.5 20.0])))
```

## BLAS Level 3 Functions

BLAS Level 3 functions are those that compute in cubic time O(n^3). They usually work with matrices and produce matrices.

```
(facts
"BLAS 3 mm: Here is how you can multiply matrices. Note that this is matrix
multiplication, NOT an element-by-element multiplication, which is a much
simpler and less useful operation."
(let [a (dge 2 3 (range 6))
b (dge 3 1 (range 3))
c (dge 2 1 [1 2])]
(mm a b) => (dge 2 1 [10 13])
(mm 1.5 a b) => (dge 2 1 [15 19.5])
a => (dge 2 3 (range 6))
b => (dge 3 1 (range 3))
(mm! 1.0 a b c) => (dge 2 1 [11 15])
c => (dge 2 1 [11 15])
(mm! 1.5 a b 2.0 c) => (dge 2 1 [37 49.5])
c => (dge 2 1 [37 49.5])))
(facts
"Some of BLAS 1 functions, such as copy!, and swp!, axpy!, also work with matrices."
(let [a (dge 2 3 (range 6))
b (dge 2 3)]
(swp! a b) => a
a => (dge 2 3)
b => (dge 2 3 (range 6))
(copy! a b) => (dge 2 3)
b => (dge 2 3)))
```

## Useful Non-BLAS Functions

While BLAS functions are the meat of linear algebra computations, there is a bunch of other stuff that we would like to do with vectors and matrices. For example, we would like to see their structure, dimensions, to see specific entries, to get subvectors or submatrices, to transpose matrices, etc. Neanderthal offers time and space efficient implementations of such operations.

```
(facts
"Miscelaneous vector functions."
(let [x (dv (range 10 20))]
(= x (dv (range 10 20))) => true
(identical? x (dv (range 10 20))) => false
(entry x 7) => 17.0
(dim x) => 10
(vctr? x) => true
(vector? x) => false
(vctr? [1 2 3]) => false
(zero x) => (dv 10)
(subvector x 2 3) => (dv 12 13 14)
x => (dv (range 10 20)))
(facts
"Miscelaneous matrix functions."
(let [a (dge 20 30 (range 600))]
(= a (dge 20 30 (range 600))) => true
(identical? a (dge 20 30 (range 600))) => false
(entry a 15 16) => 335.0
(mrows a) => 20
(ncols a) => 30
(trans (trans a)) => a
(col a 2) => (dv (range 40 60))
(row a 3) => (dv (range 3 600 20))
(matrix? a) => true
(matrix? (row a 4)) => false
(zero a) => (dge 20 30)
(submatrix a 11 12 2 3) => (dge 2 3 [251 252 271 272 291 292])
(ax 2.0 (col a 0)) => (dv (range 0 40 2))
a => (dge 20 30 (range 600))
(scal! 2.0 (row a 3)) => (row a 3)
a =not=> (dge 20 30 (range 600))))
```

Neanderthal does all these things, and does them very fast, and usually without memory copying. You have to be careful, though. Most of the time, when you extract a part of a matrix or a vector, you get a live connection to the original data. All changes that occur to the part, will also change the original. It is often useful for performance reasons, but sometimes you want to avoid it. In that case, avoid the destructive BANG functions, or copy the data to a fresh instance before using the BANG functions. The important thing is that you always have control and can explicitly choose what you need in particular case: purity or performance, or, sometimes, both.

```
(facts
"If you change the subpart, you change the original data. "
(let [x (dv (range 10 20))]
(scal! 10.0 (copy (subvector x 2 3))) => (dv 120 130 140)
x => (dv (range 10 20))
(scal! 10.0 (subvector x 2 3)) => (dv 120 130 140)
x => (dv [10 11 120 130 140 15 16 17 18 19]))))
```

**Fast** Mapping and Reducing

BLAS routines are fast and neat. However, often we need to compute the entries of our matrices and vectors in a custom way. In Clojure, we would do that with map, reduce, filter and similar functions that work the sequence. However, sequences box all numbers, and are thus orders of magnitude slower than functions working on primitive arrays. On the other hand, primitive arrays areduce and amap are macros and a bit awkward…

Fortunataly, Neanderthal comes with its own map and reduce functions that:

- Work on primitives
- Accept primitive hinted functions
- Can transform the data

This way, we get the full elegance of map and reduce with the speed (almost) as fast as looping on primitive arrays with primitive functions. See the benchmarks for performance details; here we only demonstrate how these are used.

```
(let [primitive-inc (fn ^double [^double x] (inc x))
primitive-add2 (fn ^double [^double x ^double y] (+ x y))
primitive-add3 (fn ^double [^double x ^double y ^double z] (+ x y z))
primitive-multiply (fn ^double [^double x ^double y] (* x y))
a (dge 2 3 (range 6))
b (dge 2 3 (range 0 60 10))
c (dge 2 3 (range 0 600 100))]
(fact
"You can change individual entries of any structure with fmap!. You can also
accumulate values with fold, or fold the entries."
(fmap! primitive-inc a) => (dge 2 3 (range 1 7))
a => (dge 2 3 (range 1 7))
(fmap! primitive-add3 a b c) => (dge 2 3 [1 112 223 334 445 556])
a => (dge 2 3 [1 112 223 334 445 556])
(fold primitive-add2 0.0 b c) => 1650.0
(fold c) => 1500.0
(fold primitive-multiply 1.0 a) => 2.06397368128E12))
```

## LAPACK Functions

LAPACK functions build on BLAS, they are performed on the same vector and matrix objects: dv, dge etc.

From version 0.9.0 on, Neanderthal supports major (but not yet all) LAPACK operations. For the details on
how to use them, check out the `test`

folder in the Neanderthal github repository, and
My blog at dragan.rocks.

## Can Neanderthal Go Even Faster?

Yes, it can, MUCH faster. Neanderthal have a pluggable infrastructure, and already comes with a GPU engine that can process your data on graphic cards orders of magnitude faster than on the CPU. Check out the GPU tutorial!

## Additional Examples

Neanderthal comes with additional examples from the Linear algebra literature. Check out the Examples folder for the always accurate and working examples in the form of midje tests. “

## Tell Us What You Think!

Please take some time to tell us about your experience with the library and this site. Let us know what we should be explaining or is not clear enough. If you are willing to contribute improvements, even better!