# MAT Manual

###### [in package MGL-MAT]

- [system] "mgl-mat"

    - Version: 0.1.0

    - Description: mat is library for working with multi-dimensional
    arrays which supports efficient interfacing to foreign and CUDA
    code. BLAS and CUBLAS bindings are available.

    - Licence: MIT, see COPYING.

    - Author: Gábor Melis <mega@retes.hu>

    - Mailto: [mega@retes.hu](mailto:mega@retes.hu)

    - Homepage: <http://melisgl.github.io/mgl-mat>

    - Bug tracker: <https://github.com/melisgl/mgl-mat/issues>

    - Source control: [GIT](https://github.com/melisgl/mgl-mat.git)

    - Depends on: alexandria, bordeaux-threads, cffi, cffi-grovel, cl-cuda, flexi-streams, ieee-floats, lla, mgl-pax, num-utils, static-vectors, trivial-garbage

## Links

Here is the [official
repository](https://github.com/melisgl/mgl-mat) and the [HTML
documentation](http://melisgl.github.io/mgl-mat/mat-manual.html)
for the latest version.

## Introduction

### What's MGL-MAT?

MGL-MAT is library for working with multi-dimensional arrays
which supports efficient interfacing to foreign and CUDA code with
automatic translations between cuda, foreign and lisp storage. BLAS
and CUBLAS bindings are available.

### What kind of matrices are supported?

Currently only row-major single and double float matrices are
supported, but it would be easy to add single and double precision
complex types too. Other numeric types, such as byte and native
integer, can be added too, but they are not supported by CUBLAS.
There are no restrictions on the number of dimensions, and reshaping
is possible. All functions operate on the visible portion of the
matrix (which is subject to displacement and shaping), invisible
elements are not affected.

### Where to Get it?

All dependencies are in quicklisp except for
[CL-CUDA](https://github.com/takagi/cl-cuda) that needs to be
fetched from github. Just clone CL-CUDA and MGL-MAT into
quicklisp/local-projects/ and you are set. MGL-MAT itself lives
[at github](https://github.com/melisgl/mgl-mat), too.

Prettier-than-markdown HTML documentation cross-linked with other
libraries is
[available](http://melisgl.github.io/mgl-pax-world/mat-manual.html)
as part of [PAX World](http://melisgl.github.io/mgl-pax-world/).

## Tutorial

We are going to see how to create matrices, access their contents.

Creating matrices is just like creating lisp arrays:

    (make-mat '6)
    ==> #<MAT 6 A #(0.0d0 0.0d0 0.0d0 0.0d0 0.0d0 0.0d0)>
    
    (make-mat '(2 3) :ctype :float :initial-contents '((1 2 3) (4 5 6)))
    ==> #<MAT 2x3 AB #2A((1.0 2.0 3.0) (4.0 5.0 6.0))>
    
    (make-mat '(2 3 4) :initial-element 1)
    ==> #<MAT 2x3x4 A #3A(((1.0d0 1.0d0 1.0d0 1.0d0)
    -->                    (1.0d0 1.0d0 1.0d0 1.0d0)
    -->                    (1.0d0 1.0d0 1.0d0 1.0d0))
    -->                   ((1.0d0 1.0d0 1.0d0 1.0d0)
    -->                    (1.0d0 1.0d0 1.0d0 1.0d0)
    -->                    (1.0d0 1.0d0 1.0d0 1.0d0)))>

The most prominent difference from lisp arrays is that mats are
always numeric and their element type (called ctype here) defaults
to :double.

Individual elements can be accessed or set with mref:

    (let ((m (make-mat '(2 3))))
      (setf (mref m 0 0) 1)
      (setf (mref m 0 1) (* 2 (mref m 0 0)))
      (incf (mref m 0 2) 4)
      m)
    ==> #<MAT 2x3 AB #2A((1.0d0 2.0d0 4.0d0) (0.0d0 0.0d0 0.0d0))>

Compared to aref mref is a very expensive operation and it's best
used sparingly. Instead, typical code relies much more on matrix
level operations:

    (princ (scal! 2 (fill! 3 (make-mat 4))))
    .. #<MAT 4 BF #(6.0d0 6.0d0 6.0d0 6.0d0)>
    ==> #<MAT 4 ABF #(6.0d0 6.0d0 6.0d0 6.0d0)>

The content of a matrix can be accessed in various representations
called facets. MGL-MAT takes care of synchronizing these facets
by copying data around lazily, but doing its best to share storage
for facets that allow it.

Notice the abf in the printed results. It illustrates that behind
the scenes fill! worked on the backing-array
facet (hence the b) that's basically a 1d lisp array. scal! on the
other hand made a foreign call to the BLAS dscal function for
which it needed the foreign-array facet (f).
Finally, the a stands for the array facet that was
created when the array was printed. All facets are up-to-date (else
some of the characters would be lowercase). This is possible because
these three facets actually share storage which is never the case
for the cuda-array facet. Now if we have a
CUDA-capable GPU, CUDA can be enabled with with-cuda*:

    (with-cuda* ()
      (princ (scal! 2 (fill! 3 (make-mat 4)))))
    .. #<MAT 4 C #(6.0d0 6.0d0 6.0d0 6.0d0)>
    ==> #<MAT 4 A #(6.0d0 6.0d0 6.0d0 6.0d0)>

Note the lonely c showing that only the cuda-array
facet was used for both fill! and scal!. When with-cuda* exits and
destroys the CUDA context, it destroys all CUDA facets, moving their
data to the array facet, so the returned mat only has
that facet.

When there is no high-level operation that does what we want, we may
need to add new operations. This is usually best accomplished by
accessing one of the facets directly, as in the following example:

    (defun logdet (mat)
      "Logarithm of the determinant of MAT. Return -1, 1 or 0 (or
      equivalent) to correct for the sign, as the second value."
      (with-facets ((array (mat 'array :direction :input)))
        (lla:logdet array)))

Notice that logdet doesn't know about CUDA at all. with-facets
gives it the content of the matrix as a normal multidimensional lisp
array, copying the data from the GPU or elsewhere if necessary. This
allows new representations (facets) to be added easily and it also
avoids copying if the facet is already up-to-date. Of course, adding
CUDA support to logdet could make it more efficient.

Adding support for matrices that, for instance, live on a remote
machine is thus possible with a new facet type and existing code
would continue to work (albeit possibly slowly). Then one could
optimize the bottleneck operations by sending commands over the
network instead of copying data.

It is a bad idea to conflate resource management policy and
algorithms. MGL-MAT does its best to keep them separate.

## Basics

- [class] mat cube

    A mat is a data cube that is much like a lisp
    array, it supports displacement, arbitrary dimensions and
    initial-element with the usual semantics. However, a mat supports
    different representations of the same data. See Tutorial for
    an introduction.

- [reader] mat-ctype mat (:ctype = *default-mat-ctype*)

    One of *supported-ctypes*. The matrix can hold
    only values of this type.

- [reader] mat-displacement mat (:displacement = 0)

    A value in the [0,max-size] interval. This is
    like the DISPLACED-INDEX-OFFSET of a lisp array, but displacement
    is relative to the start of the underlying storage vector.

- [reader] mat-dimensions mat (:dimensions)

    Like array-dimensions. It holds a list of
    dimensions, but it is allowed to pass in scalars too.

- [function] mat-dimension mat axis-number

    Return the dimension along axis-number. Similar to
    array-dimension.

- [reader] mat-initial-element mat (:initial-element = 0)

    If non-nil, then when a facet is created, it is
    filled with initial-element coerced to the appropriate numeric
    type. If nil, then no initialization is performed.

- [reader] mat-size mat

    The number of elements in the visible portion of
    the array. This is always the product of the elements
    mat-dimensions and is similar to array-total-size.

- [reader] mat-max-size mat (:max-size)

    The number of elements for which storage may be
    allocated. This is displacement + mat-size + slack where slack
    is the number of trailing invisible elements.

- [function] make-mat dimensions &rest args &key (ctype *default-mat-ctype*) (displacement 0) max-size initial-element initial-contents (synchronization *default-synchronization*) displaced-to (cuda-enabled *default-mat-cuda-enabled*)

    Return a new mat object. If initial-contents is given then the
    matrix contents are initialized with replace!. See class mat for the
    description of the rest of the parameters. This is exactly
    what (make-instance 'mat ...) does except dimensions is not a
    keyword argument so that make-mat looks more like make-array. The
    semantics of synchronization are desribed in the
    Synchronization section.

    If specified, displaced-to must be a mat object large enough (in the
    sense of its mat-size), to hold displacement plus (reduce #'*
    dimensions) elements. Just like with make-array, initial-element
    and initial-contents must not be supplied together with
    displaced-to. See Shaping for more.

- [function] array-to-mat array &key ctype (synchronization *default-synchronization*)

    Create a mat that's equivalent to array. Displacement of the
    created array will be 0 and the size will be equal to
    array-total-size. If ctype is non-nil, then it will be the ctype of
    the new matrix. Else array's type is converted to a ctype. If there
    is no corresponding ctype, then *default-mat-ctype* is used.
    Elements of array are coerced to ctype.

    Also see Synchronization.

- [function] mat-to-array mat

- [function] replace! mat seq-of-seqs

    Replace the contents of mat with the elements of seq-of-seqs.
    seq-of-seqs is a nested sequence of sequences similar to the
    initial-contents argument of make-array. The total number of
    elements must match the size of mat. Returns mat.

    seq-of-seqs may contain multi-dimensional arrays as leafs, so the
    following is legal:

        (replace! (make-mat '(1 2 3)) '(#2A((1 2 3) (4 5 6))))
        ==> #<MAT 1x2x3 AB #3A(((1.0d0 2.0d0 3.0d0) (4.0d0 5.0d0 6.0d0)))>

- [function] mref mat &rest indices

    Like aref for arrays. Don't use this if you care about performance
    at all. setfable. When set, the value is coerced to the ctype of mat
    with coerce-to-ctype. Note that currently mref always operates on
    the backing-array facet so it can trigger copying of facets. When
    it's setf'ed, however, it will update the cuda-array if cuda is
    enabled and it is up-to-date or there are no facets at all.

- [function] row-major-mref mat index

    Like row-major-aref for arrays. Don't use this if you care about
    performance at all. setfable. When set, the value is coerced to the
    ctype of mat with coerce-to-ctype. Note that currently
    row-major-mref always operates on the backing-array facet so it can
    trigger copying of facets. When it's setf'ed, however, it will
    update the cuda-array if cuda is enabled and it is up-to-date or
    there are no facets at all.

- [function] mat-row-major-index mat &rest subscripts

    Like array-row-major-index for arrays.

## Element types

- [variable] *supported-ctypes* (:float :double)

- [type] ctype

    This is basically (member :float :double).

- [variable] *default-mat-ctype* :double

    By default mats are created with this ctype. One of :float
    or :double.

- [function] coerce-to-ctype x &key (ctype *default-mat-ctype*)

    Coerce the scalar x to the lisp type corresponding to ctype.

## Printing

- [variable] *print-mat* t

    Controls whether the contents of a mat object are printed as an
    array (subject to the standard printer control variables).

- [variable] *print-mat-facets* t

    Controls whether a summary of existing and up-to-date facets is
    printed when a mat object is printed. The summary that looks like
    ABcfh indicates that all five facets (array,
    backing-array, cuda-array,
    foreign-array, cuda-host-array) are
    present and the first two are up-to-date. A summary of a single #-
    indicates that there are no facets.

## Shaping

We are going to discuss various ways to change the visible portion
and dimensions of matrices. Conceptually a matrix has an underlying
non-displaced storage vector. For (make-mat 10 :displacement
7 :max-size 21) this underlying vector looks like this:

    displacement | visible elements  | slack
    . . . . . . . 0 0 0 0 0 0 0 0 0 0 . . . .

Whenever a matrix is reshaped (or displaced to in lisp
terminology), its displacement and dimensions change but the
underlying vector does not.

The rules for accessing displaced matrices is the same as always:
multiple readers can run in parallel, but attempts to write will
result in an error if there are either readers or writers on any of
the matrices that share the same underlying vector.

### Comparison to Lisp Arrays

One way to reshape and displace mat objects is with make-mat and
its displaced-to argument whose semantics are similar to that of
make-array in that the displacement is relative to the
displacement of displaced-to.

    (let* ((base (make-mat 10 :initial-element 5 :displacement 1))
           (mat (make-mat 6 :displaced-to base :displacement 2)))
      (fill! 1 mat)
      (values base mat))
    ==> #<MAT 1+10+0 A #(5.0d0 5.0d0 1.0d0 1.0d0 1.0d0 1.0d0 1.0d0 1.0d0 5.0d0
    -->                  5.0d0)>
    ==> #<MAT 3+6+2 AB #(1.0d0 1.0d0 1.0d0 1.0d0 1.0d0 1.0d0)>

There are important semantic differences compared to lisp arrays all
which follow from the fact that displacement operates on the
underlying conceptual non-displaced vector.

- Matrices can be displaced and have slack even without displaced-to
  just like base in the above example.

- It's legal to alias invisible elements of displaced-to as long as
  the new matrix fits into the underlying storage.

- Negative displacements are allowed with displaced-to as long as
  the adjusted displacement is non-negative.

- Further shaping operations can make invisible portions of the
  displaced-to matrix visible by changing the displacement.

- In contrast to array-displacement, mat-displacement only returns
  an offset into the underlying storage vector.

### Functional Shaping

The following functions are collectively called the functional
shaping operations, since they don't alter their arguments in any
way. Still, since storage is aliased modification to the returned
matrix will affect the original.

- [function] reshape-and-displace mat dimensions displacement

    Return a new matrix of dimensions that aliases mat's storage at
    offset displacement. displacement 0 is equivalent to the start of
    the storage of mat regardless of mat's displacement.

- [function] reshape mat dimensions

    Return a new matrix of dimensions whose displacement is the same as
    the displacement of mat.

- [function] displace mat displacement

    Return a new matrix that aliases mat's storage at offset
    displacement. displacement 0 is equivalent to the start of the
    storage of mat regardless of mat's displacement. The returned matrix
    has the same dimensions as mat.

### Destructive Shaping

The following destructive operations don't alter the contents of
the matrix, but change what is visible. adjust! is the odd one out,
it may create a new mat.

- [function] reshape-and-displace! mat dimensions displacement

    Change the visible (or active) portion of mat by altering its
    displacement offset and dimensions. Future operations will only
    affect this visible portion as if the rest of the elements were not
    there. Return mat.

    displacement + the new size must not exceed mat-max-size.
    Furthermore, there must be no facets being viewed (with with-facets)
    when calling this function as the identity of the facets is not
    stable.

- [function] reshape! mat dimensions

    Like reshape-and-displace! but only alters the dimensions.

- [function] displace! mat displacement

    Like reshape-and-displace! but only alters the displacement.

- [function] reshape-to-row-matrix! mat row

    Reshape the 2d mat to make only a single row visible. This is made
    possible by the row-major layout, hence no column counterpart.
    Return mat.

- [macro] with-shape-and-displacement (mat &optional (dimensions nil) (displacement nil)) &body body

    Reshape and displace mat if dimensions and/or displacement is given
    and restore the original shape and displacement after body is
    executed. If neither is specificed, then nothing will be changed,
    but body is still allowed to alter the shape and displacement.

- [function] adjust! mat dimensions displacement &key (destroy-old-p t)

    Like reshape-and-displace! but creates a new matrix if mat isn't
    large enough. If a new matrix is created, the contents are not
    copied over and the old matrix is destroyed with destroy-cube if
    destroy-old-p.

## Assembling

The functions here assemble a single mat from a number of
mats.

- [function] stack! axis mats mat

    Stack mats along axis into mat and return mat. If axis is 0, place
    mats into mat below each other starting from the top. If axis is 1,
    place mats side by side starting from the left. Higher axis are also
    supported. All dimensions except for axis must be the same for all
    mats.

- [function] stack axis mats &key (ctype *default-mat-ctype*)

    Like stack! but return a new mat of ctype.

        (stack 1 (list (make-mat '(3 2) :initial-element 0)
                       (make-mat '(3 1) :initial-element 1)))
        ==> #<MAT 3x3 B #2A((0.0d0 0.0d0 1.0d0)
        -->                 (0.0d0 0.0d0 1.0d0)
        -->                 (0.0d0 0.0d0 1.0d0))>

## Caching

Allocating and initializing a mat object and its necessary facets
can be expensive. The following macros remember the previous value
of a binding in the same thread and /place/. Only weak references
are constructed so the cached objects can be garbage collected.

While the cache is global, thread safety is guaranteed by having
separate subcaches per thread. Each subcache is keyed by a /place/
object that's either explicitly specified or else is unique to each
invocation of the caching macro, so different occurrences of caching
macros in the source never share data. Still, recursion could lead
to data sharing between different invocations of the same function.
To prevent this, the cached object is removed from the cache while
it is used so other invocations will create a fresh one which isn't
particularly efficient but at least it's safe.

- [macro] with-thread-cached-mat (var dimensions &rest args &key (place :scratch) (ctype '*default-mat-ctype*) (displacement 0) max-size (initial-element 0) initial-contents) &body body

    Bind var to a matrix of dimensions, ctype, etc. Cache this matrix,
    and possibly reuse it later by reshaping it. When body exits the
    cached object is updated with the binding of var which body may
    change.

    There is a separate cache for each thread and each place (under
    eq). Since every cache holds exactly one mat per ctype, nested
    with-thread-cached-mat often want to use different places. By
    convention, these places are called :scratch-1, :scratch-2,
    etc.

- [macro] with-thread-cached-mats specs &body body

    A shorthand for writing nested with-thread-cached-mat calls.

        (with-thread-cached-mat (a ...)
          (with-thread-cached-mat (b ...)
            ...))

    is equivalent to:

        (with-thread-cached-mat ((a ...)
                                 (b ...))
          ...)

- [macro] with-ones (var dimensions &key (ctype '*default-mat-ctype*) (place :ones)) &body body

    Bind var to a matrix of dimensions whose every element is 1. The
    matrix is cached for efficiency.

## BLAS Operations

Only some BLAS functions are implemented, but it should be easy to
add more as needed. All of them default to using CUDA, if it is
initialized and enabled (see use-cuda-p).

Level 1 BLAS operations

- [function] asum x &key (n (mat-size x)) (incx 1)

    Return the l1 norm of x, that is, sum of the absolute values of its
    elements.

- [function] axpy! alpha x y &key (n (mat-size x)) (incx 1) (incy 1)

    Set y to alpha * x + y. Return y.

- [function] copy! x y &key (n (mat-size x)) (incx 1) (incy 1)

    Copy x into y. Return y.

- [function] dot x y &key (n (mat-size x)) (incx 1) (incy 1)

    Return the dot product of x and y.

- [function] nrm2 x &key (n (mat-size x)) (incx 1)

    Return the l2 norm of x, which is the square root of the sum of the
    squares of its elements.

- [function] scal! alpha x &key (n (mat-size x)) (incx 1)

    Set x to alpha * x. Return x.

Level 3 BLAS operations

- [function] gemm! alpha a b beta c &key transpose-a? transpose-b? m n k lda ldb ldc

    Basically c = alpha * a' * b' + beta * c. a' is a or its transpose
    depending on transpose-a?. b' is b or its transpose depending on
    transpose-b?. Returns c.

    a' is an MxK matrix. b' is a KxN matrix. c is an MxN matrix.

    lda is the width of the matrix a (not of a'). If a is not transposed,
    then k <= lda, if it's transposed then m <= lda.

    ldb is the width of the matrix b (not of b'). If b is not transposed,
    then n <= ldb, if it's transposed then k <= ldb.

    In the example below M=3, N=2, K=5, LDA=6, LDB=3, LDC=4. The cells
    marked with + do not feature in the calculation.

                   N
                  --+
                  --+
                K -B+
                  --+
                  --+
                  +++
            K
          -----+  --++
        M --A--+  -C++
          -----+  --++
          ++++++  ++++

## Destructive API

- [function] .square! x &key (n (mat-size x))

    Set x to its elementwise square. Return x.

- [function] .sqrt! x &key (n (mat-size x))

    Set x to its elementwise square root. Return x.

- [function] .log! x &key (n (mat-size x))

    Set x to its elementwise natural logarithm. Return x.

- [function] .exp! x &key (n (mat-size x))

    Apply exp elementwise to x in a destructive manner. Return x.

- [function] .expt! x power

    Raise matrix x to power in an elementwise manner. Return x. Note
    that CUDA and non-CUDA implementations may disagree on the treatment
    of NaNs, infinities and complex results. In particular, the lisp
    implementation always computes the realpart of the results while
    CUDA's pow() returns NaNs instead of complex numbers.

- [function] .inv! x &key (n (mat-size x))

    Set x to its elementwise inverse (/ 1 x). Return x.

- [function] .logistic! x &key (n (mat-size x))

    Destructively apply the logistic function to x in an elementwise
    manner. Return x.

- [function] .+! alpha x

    Add the scalar alpha to each element of x destructively modifying
    x. Return x.

- [function] .*! x y

- [function] geem! alpha a b beta c

    Like gemm!, but multiplication is elementwise. This is not a
    standard BLAS routine.

- [function] geerv! alpha a x beta b

    GEneric Elementwise Row - Vector multiplication. B = beta * B +
    alpha a .* X* where x* is a matrix of the same shape as a whose
    every row is x. Perform elementwise multiplication on each row of a
    with the vector x and add the scaled result to the corresponding row
    of b. Return b. This is not a standard BLAS routine.

- [function] .<! x y

    For each element of x and y set y to 1 if the element in y is
    greater than the element in x, and to 0 otherwise. Return y.

- [function] .min! alpha x

    Set each element of x to alpha if it's greater than alpha. Return
    x.

- [function] .max! alpha x

    Set each element of x to alpha if it's less than alpha. Return x.

- [function] add-sign! alpha a beta b

    Add the elementwise sign (-1, 0 or 1 for negative, zero and
    positive numbers respectively) of a times alpha to beta * b. Return
    b.

- [function] fill! alpha x &key (n (mat-size x))

    Fill matrix x with alpha. Return x.

- [function] sum! x y &key axis (alpha 1) (beta 0)

    Sum matrix x along axis and add alpha * sums to beta * y
    destructively modifying y. Return y. On a 2d matrix (nothing else is
    supported currently), if axis is 0, then columns are summed, if axis
    is 1 then rows are summed.

- [function] scale-rows! scales a &key (result a)

    Set result to diag(scales)*a and return it. a is an MxN
    matrix, scales is treated as a length m vector.

- [function] scale-columns! scales a &key (result a)

    Set result to a*diag(scales) and return it. a is an MxN
    matrix, scales is treated as a length n vector.

- [function] .sin! x &key (n (mat-size x))

    Apply sin elementwise to x in a destructive manner. Return x.

- [function] .cos! x &key (n (mat-size x))

    Apply cos elementwise to x in a destructive manner. Return x.

- [function] .tan! x &key (n (mat-size x))

    Apply tan elementwise to x in a destructive manner. Return x.

- [function] .sinh! x &key (n (mat-size x))

    Apply sinh elementwise to x in a destructive manner. Return x.

- [function] .cosh! x &key (n (mat-size x))

    Apply cosh elementwise to x in a destructive manner. Return x.

- [function] .tanh! x &key (n (mat-size x))

    Apply tanh elementwise to x in a destructive manner. Return x.

Finally, some neural network operations.

- [function] convolve! x w y &key start stride anchor batched

    y = y + conv(x, w) and return y. If batched, then the first
    dimension of x and y is the number of elements in the batch (B),
    else B is assumed to be 1. The rest of the dimensions encode the
    input (x) and output (y) N dimensional feature maps. start, stride
    and anchor are lists of length N. start is the multi-dimensional
    index of the first element of the input feature map (for each
    element in the batch) for which the convolution must be computed.
    Then (elt stride (- N 1)) is added to the last element of start and
    so on until (array-dimension x 1) is reached. Then the last element
    of start is reset, (elt stride (- N 2)) is added to the first but
    last element of start and we scan the last dimension again. Take a
    2d example, start is (0 0), stride is (1 2), and x is a B*2x7
    matrix.

    w is:

        1 2 1
        2 4 2
        1 2 1

    and anchor is (1 1) which refers to the element of w whose value is
    4. This anchor point of w is placed over elements of x whose multi
    dimensional index is in numbers in this figure (only one element in
    the batch is shown):

        0,0 . 0,2 . 0,4 . 0,6
        1,0 . 1,2 . 1,4 . 1,6

    When applying w at position P of x, the convolution is the sum of
    the products of overlapping elements of x and w when w's anchor is
    placed at P. Elements of w over the edges of x are multiplied with 0
    so are effectively ignored. The order of application of w to
    positions defined by start, stride and anchor is undefined.

    y must be a B*2x4 (or 2x4 if not batched) matrix in this example,
    just large enough to hold the results of the convolutions.

- [function] derive-convolve! x xd w wd yd &key start stride anchor batched

    Add the dF/dX to xd and and dF/dW to wd where yd is dF/dY for some
    function F where Y is the result of convolution with the same
    arguments. 

- [function] max-pool! x y &key start stride anchor batched pool-dimensions

- [function] derive-max-pool! x xd y yd &key start stride anchor batched pool-dimensions

    Add the dF/dX to xd and and dF/dW to WD where yd is dF/dY for some
    function F where y is the result of max-pool! with the same
    arguments. 

## Non-destructive API

- [function] copy-mat a

    Return a copy of the active portion with regards to displacement
    and shape of a. 

- [function] copy-row a row

    Return row of a as a new 1d matrix.

- [function] copy-column a column

    Return column of a as a new 1d matrix.

- [function] mat-as-scalar a

    Return the first element of a. a must be of size 1.

- [function] scalar-as-mat x &key (ctype (lisp->ctype (type-of x)))

    Return a matrix of one dimension and one element: x. ctype, the
    type of the matrix, defaults to the ctype corresponding to the type
    of x.

- [function] m= a b

    Check whether a and b, which must be matrices of the same size, are
    elementwise equal.

- [function] transpose a

    Return the transpose of a.

- [function] m* a b &key transpose-a? transpose-b?

    Compute op(a) * op(b). Where op is either the identity or the
    transpose operation depending on transpose-a? and transpose-b?.

- [function] mm* m &rest args

    Convenience function to multiply several matrices. 

    (mm* a b c) => a * b * c

- [function] m- a b

    Return a - b.

- [function] m+ a b

    Return a + b.

- [function] invert a

    Return the inverse of a.

- [function] logdet mat

    Logarithm of the determinant of mat. Return -1, 1 or 0 (or
    equivalent) to correct for the sign, as the second value.

## Mappings

- [function] map-concat fn mats mat &key key pass-raw-p

    Call fn with each element of mats and mat temporarily reshaped to
    the dimensions of the current element of mats and return mat. For
    the next element the displacement is increased so that there is no
    overlap.

    mats is keyed by key just like the CL sequence functions. Normally,
    fn is called with the matrix returned by key. However, if
    pass-raw-p, then the matrix returned by key is only used to
    calculate dimensions and the element of mats that was passed to key
    is passed to fn, too.

        (map-concat #'copy! (list (make-mat 2) (make-mat 4 :initial-element 1))
                    (make-mat '(2 3)))
        ==> #<MAT 2x3 AB #2A((0.0d0 0.0d0 1.0d0) (1.0d0 1.0d0 1.0d0))>

- [function] map-displacements fn mat dimensions &key (displacement-start 0) displacement-step

    Call fn with mat reshaped to dimensions, first displaced by
    displacement-start that's incremented by displacement-step each
    iteration while there are enough elements left for dimensions at the
    current displacement. Returns mat.

        (let ((mat (make-mat 14 :initial-contents '(-1 0 1 2 3
                                                    4 5 6 7
                                                    8 9 10 11 12))))
          (reshape-and-displace! mat '(4 3) 1)
          (map-displacements #'print mat 4))
        ..
        .. #<MAT 1+4+9 B #(0.0d0 1.0d0 2.0d0 3.0d0)> 
        .. #<MAT 5+4+5 B #(4.0d0 5.0d0 6.0d0 7.0d0)> 
        .. #<MAT 9+4+1 B #(8.0d0 9.0d0 10.0d0 11.0d0)> 

- [function] map-mats-into result-mat fn &rest mats

    Like cl:map-into but for mat objects. Destructively modifies
    result-mat to contain the results of applying fn to each element in
    the argument mats in turn.

## Random numbers

Unless noted these work efficiently with CUDA.

- [generic-function] copy-random-state state

    Return a copy of state be it a lisp or cuda random
    state.

- [function] uniform-random! mat &key (limit 1)

    Fill mat with random numbers sampled uniformly from the [0,LIMIT)
    interval of mat's type.

- [function] gaussian-random! mat &key (mean 0) (stddev 1)

    Fill mat with independent normally distributed random numbers with
    mean and stddev.

- [function] mv-gaussian-random &key means covariances

    Return a column vector of samples from the multivariate normal
    distribution defined by means (Nx1) and covariances (NxN). No CUDA
    implementation.

- [function] orthogonal-random! m &key (scale 1)

    Fill the matrix m with random values in such a way that m^t * m
    is the identity matrix (or something close if m is wide). Return m.

## I/O

- [variable] *mat-headers* t

    If true, a header with mat-ctype and mat-size is written by
    write-mat before the contents and read-mat checks that these match
    the matrix into which it is reading.

- [generic-function] write-mat mat stream

    Write mat to binary stream in portable binary
    format. Return mat. Displacement and size are taken into account,
    only visible elements are written. Also see *mat-headers*.

- [generic-function] read-mat mat stream

    Destructively modify the visible portion (with
    regards to displacement and shape) of mat by reading mat-size number
    of elements from binary stream. Return mat. Also see
    *mat-headers*.

## Debugging

The largest class of bugs has to do with synchronization of facets
being broken. This is almost always caused by an operation that
mispecifies the direction argument of with-facet. For example, the
matrix argument of scal! should be accessed with direciton :io. But
if it's :input instead, then subsequent access to the array facet
will not see the changes made by axpy!, and if it's :output, then
any changes made to the array facet since the last update of the
cuda-array facet will not be copied and from the wrong input scal!
will compute the wrong result.

Using the SLIME inspector or trying to access the
cuda-array facet from threads other than the one in
which the corresponding CUDA context was initialized will fail. For
now, the easy way out is to debug the code with CUDA disabled (see
*cuda-enabled*).

Another thing that tends to come up is figuring out where memory is
used.

- [function] mat-room &key (stream *standard-output*) (verbose t)

    Calls foreign-room and cuda-room.

- [macro] with-mat-counters (&key count n-bytes) &body body

    Count all mat allocations and also the number of bytes they may
    require. May require here really means an upper bound,
    because (make-mat (expt 2 60)) doesn't actually uses memory until
    one of its facets is accessed (don't simply evaluate it though,
    printing the result will access the array facet if
    *print-mat*). Also, while facets today all require the same number
    of bytes, this may change in the future. This is a debugging tool,
    don't use it in production.

        (with-mat-counters (:count count :n-bytes n-bytes)
          (assert (= count 0))
          (assert (= n-bytes 0))
          (make-mat '(2 3) :ctype :double)
          (assert (= count 1))
          (assert (= n-bytes (* 2 3 8)))
          (with-mat-counters (:n-bytes n-bytes-1 :count count-1)
            (make-mat '7 :ctype :float)
            (assert (= count-1 1))
            (assert (= n-bytes-1 (* 7 4))))
          (assert (= n-bytes (+ (* 2 3 8) (* 7 4))))
          (assert (= count 2)))

## Facet API

### Facets

A mat is a cube (see Cube Manual) whose facets are
different representations of numeric arrays. These facets can be
accessed with with-facets with one of the following
facet-name locatives:

- [facet-name] backing-array

    The corresponding facet's value is a one dimensional lisp array or
    a static vector that also looks exactly like a lisp array but is
    allocated in foreign memory. See *foreign-array-strategy*.

- [facet-name] array

    Same as backing-array if the matrix is one-dimensional, all
    elements are visible (see Shaping), else it's a lisp array
    displaced to the backing array.

- [facet-name] foreign-array

    The facet's value is a foreign-array which is an
    offset-pointer wrapping a CFFI pointer. See
    *foreign-array-strategy*.

- [facet-name] cuda-host-array

    This facet's value is a basically the same as that of
    foreign-array. In fact, they share storage. The
    difference is that accessing cuda-host-array ensures
    that the foreign memory region is page-locked and registered with
    the CUDA Driver API function cuMemHostRegister(). Copying between
    GPU memory (cuda-array) and registered memory is
    significantly faster than with non-registered memory and also allows
    overlapping copying with computation. See
    with-syncing-cuda-facets.

- [facet-name] cuda-array

    The facet's value is a cuda-array, which is an offset-pointer
    wrapping a cl-cuda.driver-api:cu-device-ptr, allocated with
    cl-cuda.driver-api:cu-mem-alloc and freed automatically.

Facets bound by with with-facets are to be treated as dynamic
extent: it is not allowed to keep a reference to them beyond the
dynamic scope of with-facets.

For example, to implement the fill! operation using only the
backing-array, one could do this:

    (let ((displacement (mat-displacement x))
          (size (mat-size x)))
     (with-facets ((x* (x 'backing-array :direction :output)))
       (fill x* 1 :start displacement :end (+ displacement size))))

direction is :output because we clobber all values in x. Armed
with this knowledge about the direction, with-facets will not copy
data from another facet if the backing array is not up-to-date.

To transpose a 2d matrix with the array facet:

    (destructuring-bind (n-rows n-columns) (mat-dimensions x)
      (with-facets ((x* (x 'array :direction :io)))
        (dotimes (row n-rows)
          (dotimes (column n-columns)
            (setf (aref x* row column) (aref x* column row))))))

Note that direction is :io, because we need the data in this facet
to be up-to-date (that's the input part) and we are invalidating all
other facets by changing values (that's the output part).

To sum the values of a matrix using the foreign-array
facet:

    (let ((sum 0))
      (with-facets ((x* (x 'foreign-array :direction :input)))
        (let ((pointer (offset-pointer x*)))
          (loop for index below (mat-size x)
                do (incf sum (cffi:mem-aref pointer (mat-ctype x) index)))))
      sum)

See direction for a complete description of :input, :output and :io.
For mat objects, that needs to be refined. If a mat is reshaped
and/or displaced in a way that not all elements are visible then
those elements are always kept intact and copied around. This is
accomplished by turning :output into :io automatically on such mats.

We have finished our introduction to the various facets. It must be
said though that one can do anything without ever accessing a facet
directly or even being aware of them as most operations on mats
take care of choosing the most appropriate facet behind the scenes.
In particular, most operations automatically use CUDA, if available
and initialized. See with-cuda* for detail.

### Foreign arrays

One facet of mat objects is foreign-array which is
backed by a memory area that can be a pinned lisp array or is
allocated in foreign memory depending on *foreign-array-strategy*.

- [class] foreign-array

    foreign-array wraps a foreign pointer (in
    the sense of cffi:pointerp). That is, both offset-pointer and
    base-pointer return a foreign pointer. There are no other public
    operations that work with foreign-array objects, their sole
    purpose is represent facets of mat objects.

- [variable] *foreign-array-strategy* "-see below-"

    One of :pinned, :static and :cuda-host (see type
    foreign-array-strategy). This variable controls how foreign arrays
    are handled, and it can be changed at any time.

    If it's :pinned (only supported if pinning-supported-p), then no
    separate storage is allocated for the foreign array. Instead, it
    aliases the lisp array (via the backing-array facet).

    If it's :static, then the lisp backing arrays are allocated
    statically via the static-vectors library. On some implementations,
    explicit freeing of static vectors is necessary, this is taken care
    of by finalizers or can be controlled with with-facet-barrier.
    destroy-cube and destroy-facet may also be of help.

    :cuda-host is the same as :static, but any copies to/from the
    GPU (i.e. the cuda-array facet) will be done via the
    cuda-host-array facet whose memory pages will also be
    locked and registered with cuMemHostRegister which allows quicker
    and asynchronous copying to and from CUDA land.

    The default is :pinned if available, because it's the most
    efficient. If pinning is not available, then it's :static.

- [type] foreign-array-strategy

    One of :pinned, :static and :cuda-host. See
    *foreign-array-strategy* for their semantics.

- [function] pinning-supported-p

    Return true iff the lisp implementation efficiently supports
    pinning lisp arrays. Pinning ensures that the garbage collector
    doesn't move the array in memory. Currently this is only supported on
    SBCL gencgc platforms.

- [function] foreign-room &key (stream *standard-output*) (verbose t)

    Print a summary of foreign memory usage to stream. If verbose, make
    the output human easily readable, else try to present it in a very
    concise way. Sample output with verbose:

        Foreign memory usage:
        foreign arrays: 450 (used bytes: 3,386,295,808)

    The same data presented with verbose false:

        f: 450 (3,386,295,808)

### CUDA

- [function] cuda-available-p &key (device-id 0)

    Check that a cuda context is already in initialized in the current
    thread or a device with device-id is available.

- [macro] with-cuda* (&key (enabled '*cuda-enabled*) (device-id '*cuda-default-device-id*) (random-seed '*cuda-default-random-seed*) (n-random-states '*cuda-default-n-random-states*) n-pool-bytes) &body body

    Initializes CUDA with with all bells and whistles before body and
    deinitializes it after. Simply wrapping with-cuda* around a piece
    code is enough to make use of the first available CUDA device or
    fall back on blas and lisp kernels if there is none.

    If CUDA is already initialized, then it sets up a facet barrier
    which destroys cuda-array and cuda-host-array facets after ensuring
    that the array facet is up-to-date.

    Else, if CUDA is available and enabled, then in addition to the
    facet barrier, a CUDA context is set up, *n-memcpy-host-to-device*,
    *n-memcpy-device-to-host* are bound to zero, a cublas handle
    created, and *curand-state* is bound to a curand-xorwow-state with
    n-random-states, seeded with random-seed, and allocation of device
    memory is limited to n-pool-bytes (nil means no limit, see
    CUDA Memory Management).

    Else - that is, if CUDA is not available, body is simply executed.

- [function] call-with-cuda fn &key ((:enabled *cuda-enabled*) *cuda-enabled*) (device-id *cuda-default-device-id*) (random-seed *cuda-default-random-seed*) (n-random-states *cuda-default-n-random-states*) n-pool-bytes

    Like with-cuda*, but takes a no argument function instead of the
    macro's body.

- [variable] *cuda-enabled* t

    Set or bind this to false to disable all use of cuda. If this is
    done from within with-cuda, then cuda becomes temporarily disabled.
    If this is done from outside with-cuda, then it changes the default
    values of the enabled argument of any future with-cuda*s which
    turns off cuda initialization entirely.

- [accessor] cuda-enabled mat (:cuda-enabled = *default-mat-cuda-enabled*)

    The control provided by *cuda-enabled* can be too
    coarse. This flag provides a per-object mechanism to turn cuda
    off. If it is set to nil, then any operation that pays attention
    to this flag will not create or access the cuda-array facet.
    Implementationally speaking, this is easily accomplished by using
    use-cuda-p.

- [variable] *default-mat-cuda-enabled* t

    The default for cuda-enabled.

- [variable] *n-memcpy-host-to-device* 0

    Incremented each time a host to device copy is performed. Bound to
    0 by with-cuda*. Useful for tracking down performance problems.

- [variable] *n-memcpy-device-to-host* 0

    Incremented each time a device to host copy is performed. Bound to
    0 by with-cuda*. Useful for tracking down performance problems.

- [variable] *cuda-default-device-id* 0

    The default value of with-cuda*'s :device-id argument.

- [variable] *cuda-default-random-seed* 1234

    The default value of with-cuda*'s :random-seed argument.

- [variable] *cuda-default-n-random-states* 4096

    The default value of with-cuda*'s :n-random-states argument.

#### CUDA Memory Management

The GPU (called device in CUDA terminology) has its own memory
and it can only perform computation on data in this device memory
so there is some copying involved to and from main memory. Efficient
algorithms often allocate device memory up front and minimize the
amount of copying that has to be done by computing as much as
possible on the GPU.

MGL-MAT reduces the cost of device of memory allocations by
maintaining a cache of currently unused allocations from which it
first tries to satisfy allocation requests. The total size of all
the allocated device memory regions (be they in use or currently
unused but cached) is never more than n-pool-bytes as specified in
with-cuda*. n-pool-bytes being nil means no limit.

- [condition] cuda-out-of-memory storage-condition

    If an allocation request cannot be
    satisfied (either because of n-pool-bytes or physical device memory
    limits being reached), then cuda-out-of-memory is signalled.

- [function] cuda-room &key (stream *standard-output*) (verbose t)

    When CUDA is in use (see use-cuda-p), print a summary of memory
    usage in the current CUDA context to stream. If verbose, make the
    output human easily readable, else try to present it in a very
    concise way. Sample output with verbose:

        CUDA memory usage:
        device arrays: 450 (used bytes: 3,386,295,808, pooled bytes: 1,816,657,920)
        host arrays: 14640 (used bytes: 17,380,147,200)
        host->device copies: 154,102,488, device->host copies: 117,136,434

    The same data presented with verbose false:

        d: 450 (3,386,295,808 + 1,816,657,920), h: 14640 (17,380,147,200)
        h->d: 154,102,488, d->h: 117,136,434

That's it about reducing the cost allocations. The other important
performance consideration, minimizing the amount copying done, is
very hard to do if the data doesn't fit in device memory which is
often a very limited resource. In this case the next best thing is
to do the copying concurrently with computation.

- [macro] with-syncing-cuda-facets (mats-to-cuda mats-to-cuda-host &key (safep '*syncing-cuda-facets-safe-p*)) &body body

    Update CUDA facets in a possibly asynchronous way while body
    executes. Behind the scenes, a separate CUDA stream is used to copy
    between registered host memory and device memory. When
    with-syncing-cuda-facets finishes either by returning normally or by
    a performing a non-local-exit the following are true:

    - All mats in mats-to-cuda have an up-to-date
      cuda-array facet.

    - All mats in mats-to-cuda-host have an up-to-date
      cuda-host-array facet and no
      cuda-array.

    It is an error if the same matrix appears in both mats-to-cuda and
    mats-to-cuda-host, but the same matrix may appear any number of
    times in one of them.

    If safep is true, then the all matrices in either of the two lists
    are effectively locked for output until with-syncing-cuda-facets
    finishes. With SAFE nil, unsafe accesses to facets of these matrices
    are not detected, but the whole operation has a bit less overhead.

- [variable] *syncing-cuda-facets-safe-p* t

    The default value of the safep argument of
    with-syncing-cuda-facets.

Also note that often the easiest thing to do is to prevent the use
of CUDA (and consequently the creation of cuda-array
facets, and allocations). This can be done either by binding
*cuda-enabled* to nil or by setting cuda-enabled to nil on specific
matrices.

## Writing Extensions

New operations are usually implemented in lisp, CUDA, or by calling
a foreign function in, for instance, BLAS, CUBLAS, CURAND.

### Lisp Extensions

- [macro] define-lisp-kernel (name &key (ctypes '(:float :double))) (&rest params) &body body

    This is very much like define-cuda-kernel but for normal lisp code.
    It knows how to deal with mat objects and can define the same
    function for multiple ctypes. Example:

        (define-lisp-kernel (lisp-.+!)
            ((alpha single-float) (x :mat :input) (start-x index) (n index))
          (loop for xi of-type index upfrom start-x
                  below (the! index (+ start-x n))
                do (incf (aref x xi) alpha)))

    Parameters are either of the form (<name> <lisp-type)
    or (<name> :mat <direction>). In the latter case, the appropriate
    CFFI pointer is passed to the kernel. <direction> is passed on to
    the with-facet that's used to acquire the foreign array. Note that
    the return type is not declared.

    Both the signature and the body are written as if for single floats,
    but one function is defined for each ctype in ctypes by transforming
    types, constants and code by substituting them with their ctype
    equivalents. Currently this means that one needs to write only one
    kernel for single-float and double-float. All such functions get the
    declaration from *default-lisp-kernel-declarations*.

    Finally, a dispatcher function with name is defined which determines
    the ctype of the mat objects passed for :mat typed parameters. It's
    an error if they are not of the same type. Scalars declared
    single-float are coerced to that type and the appropriate kernel is
    called.

- [variable] *default-lisp-kernel-declarations* ((optimize speed (sb-c:insert-array-bounds-checks 0)))

    These declarations are added automatically to kernel functions.

### CUDA Extensions

- [function] use-cuda-p &rest mats

    Return true if cuda is enabled (*cuda-enabled*), it's initialized
    and all mats have cuda-enabled. Operations of
    matrices use this to decide whether to go for the CUDA
    implementation or BLAS/Lisp. It's provided for implementing new
    operations.

- [function] choose-1d-block-and-grid n max-n-warps-per-block

    Return two values, one suitable as the :block-dim, the other as
    the :grid-dim argument for a cuda kernel call where both are
    one-dimensional (only the first element may be different from 1).

    The number of threads in a block is a multiple of *cuda-warp-size*.
    The number of blocks is between 1 and and *cuda-max-n-blocks*. This
    means that the kernel must be able handle any number of elements in
    each thread. For example, a strided kernel that adds a constant to
    each element of a length n vector looks like this:

        (let ((stride (* block-dim-x grid-dim-x)))
          (do ((i (+ (* block-dim-x block-idx-x) thread-idx-x)
                  (+ i stride)))
              ((>= i n))
            (set (aref x i) (+ (aref x i) alpha))))

    It is often the most efficient to have max-n-warps-per-block around
    4. Note that the maximum number of threads per block is limited by
    hardware (512 for compute capability < 2.0, 1024 for later
    versions), so *cuda-max-n-blocks* times max-n-warps-per-block must
    not exceed that limit.

- [function] choose-2d-block-and-grid dimensions max-n-warps-per-block

    Return two values, one suitable as the :block-dim, the other as
    the :grid-dim argument for a cuda kernel call where both are
    two-dimensional (only the first two elements may be different from
    1).

    The number of threads in a block is a multiple of *cuda-warp-size*.
    The number of blocks is between 1 and and *cuda-max-n-blocks*.
    Currently - but this may change - the block-dim-x is always
    *cuda-warp-size* and grid-dim-x is always 1.

    This means that the kernel must be able handle any number of
    elements in each thread. For example, a strided kernel that adds a
    constant to each element of a HEIGHT*WIDTH matrix looks like this:

        (let ((id-x (+ (* block-dim-x block-idx-x) thread-idx-x))
              (id-y (+ (* block-dim-y block-idx-y) thread-idx-y))
              (stride-x (* block-dim-x grid-dim-x))
              (stride-y (* block-dim-y grid-dim-y)))
          (do ((row id-y (+ row stride-y)))
              ((>= row height))
            (let ((i (* row width)))
              (do ((column id-x (+ column stride-x)))
                  ((>= column width))
                (set (aref x i) (+ (aref x i) alpha))
                (incf i stride-x)))))

- [function] choose-3d-block-and-grid dimensions max-n-warps-per-block

    Return two values, one suitable as the :block-dim, the other as
    the :grid-dim argument for a cuda kernel call where both are
    two-dimensional (only the first two elements may be different from
    1).

    The number of threads in a block is a multiple of *cuda-warp-size*.
    The number of blocks is between 1 and and *cuda-max-n-blocks*.
    Currently - but this may change - the block-dim-x is always
    *cuda-warp-size* and grid-dim-x is always 1.

    This means that the kernel must be able handle any number of
    elements in each thread. For example, a strided kernel that adds a
    constant to each element of a thickness * height * width 3d array
    looks like this:

        (let ((id-x (+ (* block-dim-x block-idx-x) thread-idx-x))
              (id-y (+ (* block-dim-y block-idx-y) thread-idx-y))
              (id-z (+ (* block-dim-z block-idx-z) thread-idx-z))
              (stride-x (* block-dim-x grid-dim-x))
              (stride-y (* block-dim-y grid-dim-y))
              (stride-z (* block-dim-z grid-dim-z)))
          (do ((plane id-z (+ plane stride-z)))
              ((>= plane thickness))
            (do ((row id-y (+ row stride-y)))
                ((>= row height))
              (let ((i (* (+ (* plane height) row)
                          width)))
                (do ((column id-x (+ column stride-x)))
                    ((>= column width))
                  (set (aref x i) (+ (aref x i) alpha))
                  (incf i stride-x))))))

- [macro] define-cuda-kernel (name &key (ctypes '(:float :double))) (return-type params) &body body

    This is an extended cl-cuda:defkernel macro. It knows how to deal
    with mat objects and can define the same function for multiple
    ctypes. Example:

        (define-cuda-kernel (cuda-.+!)
            (void ((alpha float) (x :mat :input) (n int)))
          (let ((stride (* block-dim-x grid-dim-x)))
            (do ((i (+ (* block-dim-x block-idx-x) thread-idx-x)
                    (+ i stride)))
                ((>= i n))
              (set (aref x i) (+ (aref x i) alpha)))))

    The signature looks pretty much like in cl-cuda:defkernel, but
    parameters can take the form of (<name> :mat <direction>) too, in
    which case the appropriate cl-cuda.driver-api:cu-device-ptr is
    passed to the kernel. <direction> is passed on to the with-facet
    that's used to acquire the cuda array.

    Both the signature and the body are written as if for single floats,
    but one function is defined for each ctype in ctypes by transforming
    types, constants and code by substituting them with their ctype
    equivalents. Currently this means that one needs to write only one
    kernel for float and double.

    Finally, a dispatcher function with name is defined which determines
    the ctype of the mat objects passed for :mat typed parameters. It's
    an error if they are not of the same type. Scalars declared
    float are coerced to that type and the appropriate
    kernel is called.

#### CUBLAS

In a with-cuda* BLAS Operations will automatically use CUBLAS. No need to
use these at all.

- [condition] cublas-error error

- [reader] cublas-error-function-name cublas-error (:function-name)

- [reader] cublas-error-status cublas-error (:status)

- [variable] *cublas-handle*

- [function] cublas-create handle

- [function] cublas-destroy &key (handle *cublas-handle*)

- [macro] with-cublas-handle nil &body body

- [function] cublas-get-version version &key (handle *cublas-handle*)

#### CURAND

This the low level CURAND API. You probably want Random numbers
instead.

- [macro] with-curand-state (state) &body body

- [variable] *curand-state*

- [class] curand-xorwow-state

- [reader] n-states curand-xorwow-state (:n-states)

- [reader] states curand-xorwow-state (:states)