Awesome

MAT Manual

1 The MGL-MAT ASDF System
2 Links
3 Introduction
4 Tutorial
5 Basics
6 Element types
7 Printing
8 Shaping
9 Assembling
10 Caching
11 BLAS Operations
12 Destructive API
13 Non-destructive API
14 Mappings
15 Random numbers
16 I/O
17 Debugging
18 Facet API
19 Writing Extensions
- 19.1 Lisp Extensions
- 19.2 CUDA Extensions
  - 19.2.1 CUBLAS
  - 19.2.2 CURAND

[in package MGL-MAT]

1 The `MGL-MAT` ASDF System

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
Homepage: http://melisgl.github.io/mgl-mat
Bug tracker: https://github.com/melisgl/mgl-mat/issues
Source control: GIT

2 Links

Here is the official repository and the HTML documentation for the latest version.

3 Introduction

3.1 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.

3.2 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.

3.3 Where to Get it?

All dependencies are in quicklisp except for 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, too.

Prettier-than-markdown HTML documentation cross-linked with other libraries is available as part of PAX World.

4 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(0 1) 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.

5 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(0 1). 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(0 1) 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.

6 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.

7 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.

8 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.

8.1 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.

8.2 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.

8.3 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(0 1)) 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.

9 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))>

10 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.

11 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++
    -----+  --++
    ++++++  ++++
```

12 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.

13 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.

14 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.

15 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.

16 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*.

17 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(0 1) 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)))
```

18 Facet API

18.1 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(0 1) 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(0 1) 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.

18.2 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)
```

18.3 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.

18.3.1 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.

19 Writing Extensions

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

19.1 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.

19.2 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.

19.2.1 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*)

19.2.2 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) <a id="x-28MGL-CUBE-3A-40CUBE-MANUAL-20MGL-PAX-3ASECTION-29"></a>

Cube Manual

1 Links
2 Introduction
3 Basics
4 Synchronization
5 Facets
6 Facet Extension API
7 The Default Implementation of CALL-WITH-FACET*
8 Lifetime
- 8.1 Facet Barriers

[in package MGL-CUBE]

1 Links

Here is the official repository and the HTML documentation for the latest version.

2 Introduction

This is the library on which MGL-MAT (see MAT Manual) is built. The idea of automatically translating between various representations may be useful for other applications, so this got its own package and all ties to MGL-MAT has been severed.

This package defines CUBE, an abstract base class that provides a framework for automatic conversion between various representations of the same data. To define a cube, CUBE needs to be subclassed and the Facet Extension API be implemented.

If you are only interested in how to use cubes in general, read Basics, Lifetime and Facet Barriers.

If you want to implement a new cube datatype, then see Facets, Facet Extension API, and The Default Implementation of CALL-WITH-FACET*.

3 Basics

Here we learn what a CUBE is and how to access the data in it with WITH-FACET.

[class] CUBE

A datacube that has various representations of the same stuff. These representations go by the name `facet'. Clients must use WITH-FACET to acquire a dynamic extent reference to a facet. With the information provided in the DIRECTION argument of WITH-FACET, the cube keeps track of which facets are up-to-date and copies data between them as necessary.

The cube is an abstract class, it does not provide useful behavior in itself. One must subclass it and implement the Facet Extension API.

Also see Lifetime and Facet Barriers.

[macro] WITH-FACET (VAR (CUBE FACET-NAME &KEY (DIRECTION :IO) TYPE)) &BODY BODY

Find or create the facet with FACET-NAME in CUBE and bind VAR to the representation of CUBE's data provided by that facet. This representation is called the facet's value. The value is to be treated as dynamic extent: it is not allowed to keep a reference to it. For the description of the DIRECTION parameter, see the type DIRECTION.

If TYPE is specified, then VAR is declared to be of that type.

[type] DIRECTION

Used by WITH-FACET, DIRECTION can be :INPUT, :OUTPUT or :IO.
- :INPUT promises that the facet will only be read and never written. Other up-to-date facets of the same cube remain up-to-date. If the facet in question is not up-to-date then data is copied to it from one of the up-to-date facets (see SELECT-COPY-SOURCE-FOR-FACET*).
- :OUTPUT promises that all data will be overwritten without reading any data. All up-to-date facets become non-up-to-date, while this facet is marked as up-to-date. No copying of data takes place.
- :IO promises nothing about the type of access. All up-to-date facets become non-up-to-date, while this facet is marked as up-to-date. If the facet in question is not up-to-date then data is copied to it from one of the up-to-date facets (see SELECT-COPY-SOURCE-FOR-FACET*).
Any number of WITH-FACETs with direction :INPUT may be active at the same time, but :IO and :OUTPUT cannot coexists with another WITH-FACET regardless of the direction. The exception for this rule is that an inner WITH-FACET does not conflict with an enclosing WITH-FACET if they are for the same facet (but inner WITH-FACETs for another facet or for the same facet from another thread do).

See CHECK-NO-WRITERS and CHECK-NO-WATCHERS called by The Default Implementation of CALL-WITH-FACET*.

[macro] WITH-FACETS (&REST FACET-BINDING-SPECS) &BODY BODY

A shorthand for writing nested WITH-FACET calls.

(with-facet (f1 (c1 'name1 :direction :input))
  (with-facet (f2 (c2 'name2 :direction :output))
    ...))

is equivalent to:

(with-facets ((f1 (c1 'name1 :direction :input))
              (f2 (c2 'name2 :direction :output)))
  ...)

4 Synchronization

Cubes keep track of which facets are used, which are up-to-date to be able to perform automatic translation between facets. WITH-FACET and other operations access and make changes to this metadata so thread safety is a concern. In this section, we detail how to relax the default thread safety guarantees.

A related concern is async signal safety which arises most often when C-c'ing or killing a thread or when the extremely nasty WITH-TIMEOUT macro is used. In a nutshell, changes to cube metadata are always made with interrupts disabled so things should be async signal safe.

[accessor] SYNCHRONIZATION CUBE (:SYNCHRONIZATION = *DEFAULT-SYNCHRONIZATION*)

By default, setup and teardown of facets by WITH-FACET is performed in a thread safe way. Corrupting internal data structures of cubes is not fun, but in the name of performance, synchronization can be turned off either dynamically or on a per instance basis.

If T, then access to cube metadata is always synchronized. If NIL, then never. If :MAYBE, then whether access is synchronized is determined by *MAYBE-SYNCHRONIZE-CUBE* that's true by default.

The default is the value of *DEFAULT-SYNCHRONIZATION* that's :MAYBE by default.

Note that the body of a WITH-FACET is never synchronized with anyone, apart from the implicit reader/writer conflict (see DIRECTION).

[variable] *DEFAULT-SYNCHRONIZATION* :MAYBE

The default value for SYNCHRONIZATION of new cubes.

[variable] *MAYBE-SYNCHRONIZE-CUBE* T

Determines whether access the cube metadata is synchronized for cubes with SYNCHRONIZATION :MAYBE.

5 Facets

The basic currency for implementing new cube types is the FACET. Simply using a cube only involves facet names and values, never facets themselves.

[function] FACETS CUBE

Return the facets of CUBE.

[function] FIND-FACET CUBE FACET-NAME

Return the facet of CUBE for the facet with FACET-NAME or NIL if no such facet exists.

[class] FACET STRUCTURE-OBJECT

A cube has facets, as we discussed in Basics. Facets holds the data in a particular representation, this is called the value of the facet. A facet holds one such value and some metadata pertaining to it: its FACET-NAME(0 1), whether it's up-to-date (FACET-UP-TO-DATE-P), etc. FACET objects are never seen when simply using a cube, they are for implementing the Facet Extension API.

[structure-accessor] FACET-NAME

A symbol that uniquely identifies the facet within a cube.

[structure-accessor] FACET-VALUE

This is what's normally exposed by WITH-FACET.

[structure-accessor] FACET-DESCRIPTION

Returned by MAKE-FACET* as its second value, this is an arbitrary object in which additional information can be stored.

[structure-accessor] FACET-UP-TO-DATE-P

Whether the cube has changed since this facet has been last updated. See FACET-UP-TO-DATE-P*.

[structure-accessor] FACET-N-WATCHERS

The number of active WITH-FACETs. Updated by WATCH-FACET and UNWATCH-FACET.

[structure-accessor] FACET-WATCHER-THREADS

The threads (one for each watcher) that have active WITH-FACETs.

[structure-accessor] FACET-DIRECTION

The direction of the last WITH-FACET on this facet.

6 Facet Extension API

Many of the generic functions in this section take FACET arguments. FACET is a structure and is not intended to be subclassed. To be able to add specialized methods, the name of the facet (FACET-NAME) is also passed as the argument right in front of the corresponding facet argument.

In summary, define EQL(0 1) specializers on facet name arguments, and use FACET-DESCRIPTION to associate arbitrary information with facets.

[generic-function] MAKE-FACET* CUBE FACET-NAME

Called by WITH-FACET (or more directly WATCH-FACET) when there is no facet with FACET-NAME. As the first value, return a new object capable of storing CUBE's data in the facet with FACET-NAME. As the second value, return a facet description which will be available as FACET-DESCRIPTION. As the third value, return a generalized boolean indicating whether this facet must be explicitly destroyed (in which case a finalizer will be added to CUBE).

[generic-function] DESTROY-FACET* FACET-NAME FACET

Free the resources associated with FACET with FACET-NAME. The cube this facet belongs to is not among the parameters because this method can be called from a finalizer on the cube (so we can't have a reference to the cube portably) which also means that it may run in an unpredictable thread.

[generic-function] COPY-FACET* CUBE FROM-FACET-NAME FROM-FACET TO-FACET-NAME TO-FACET

Copy the CUBE's data from FROM-FACET with FROM-FACET-NAME to TO-FACET with TO-FACET-NAME. Called by WITH-FACET (or more directly WATCH-FACET) when necessary. FROM-FACET is what SELECT-COPY-SOURCE-FOR-FACET* returned.

[generic-function] CALL-WITH-FACET* CUBE FACET-NAME DIRECTION FN

Call FN with an up-to-date FACET-VALUE that belongs to FACET-NAME of CUBE. WITH-FACET is directly implemented in terms of this function. See The Default Implementation of CALL-WITH-FACET* for the gory details.

Specializations will most likely want to call the default implementation (with CALL-NEXT-METHOD) but with a lambda that transforms FACET-VALUE before passing it on to FN.

[generic-function] FACET-UP-TO-DATE-P* CUBE FACET-NAME FACET

Check if FACET with FACET-NAME has been updated since the latest change to CUBE (that is, since the access to other facets with DIRECTION of :IO or :OUTPUT). The default method simply calls FACET-UP-TO-DATE-P on FACET.

One reason to specialize this is when some facets actually share common storage, so updating one make the other up-to-date as well.

[generic-function] SELECT-COPY-SOURCE-FOR-FACET* CUBE TO-NAME TO-FACET

Called when TO-FACET with TO-NAME is about to be updated by copying data from an up-to-date facet. Return the facet (or its name) from which data shall be copied. Note that if the returned facet is not FACET-UP-TO-DATE-P, then it will be updated first and another SELECT-COPY-SOURCE-FOR-FACET will take place, so be careful not to get into endless recursion. The default method simply returns the first up-to-date facet.

PAX integration follows, don't worry about it if you don't use PAX, but you really should (see MGL-PAX::@PAX-MANUAL).

[locative] FACET-NAME

The FACET-NAME locative is to refer to stuff defined with DEFINE-FACET-NAME.

[macro] DEFINE-FACET-NAME SYMBOL LAMBDA-LIST &BODY DOCSTRING

Just a macro to document that SYMBOL refers to a facet name (as in the FACET-NAME). This is totally confusing, so here is an example of how MGL-MAT (see MAT Manual) documents the MGL-MAT:BACKING-ARRAY facet:
```
(define-facet-name backing-array ()
  "The corresponding facet is a one dimensional lisp array.")
```
Which makes it possible to refer to this definition (refer as in link and M-. to) MGL-MAT:BACKING-ARRAY facet-name. See MGL-PAX::@PAX-MANUAL for more.

Also see The Default Implementation of CALL-WITH-FACET*.

7 The Default Implementation of `CALL-WITH-FACET*`

[method] CALL-WITH-FACET* (CUBE CUBE) FACET-NAME DIRECTION FN

The default implementation of CALL-WITH-FACET* is defined in terms of the WATCH-FACET and the UNWATCH-FACET generic functions. These can be considered part of the Facet Extension API.

[generic-function] WATCH-FACET CUBE FACET-NAME DIRECTION

This is what the default CALL-WITH-FACET* method, in terms of which WITH-FACET is implemented, calls first. The default method takes care of creating facets, copying and tracking up-to-dateness.

Calls CHECK-NO-WRITERS (unless *LET-INPUT-THROUGH-P*) and CHECK-NO-WATCHERS (unless *LET-OUTPUT-THROUGH-P*) depending on DIRECTION to detect situations with a writer being concurrent to readers/writers because that would screw up the tracking of up-to-dateness.

The default implementation should suffice most of the time. MGL-MAT specializes it to override the DIRECTION arg, if it's :OUTPUT but not all elements are visible due to reshaping, so that invisible elements are still copied over.

[generic-function] UNWATCH-FACET CUBE FACET-NAME

This is what the default CALL-WITH-FACET* method, in terms of which WITH-FACET is implemented, calls last. The default method takes care of taking down facets. External resource managers may want to hook into this to handle unused facets.

[variable] *LET-INPUT-THROUGH-P* NIL

If true, WITH-FACETS(0 1) (more precisely, the default implementation of CALL-WITH-FACET*) with :DIRECTION :INPUT does not call CHECK-NO-WRITERS. This knob is intended to be bound locally for debugging purposes.

[variable] *LET-OUTPUT-THROUGH-P* NIL

If true, WITH-FACETS(0 1) (more precisely, the default implementation of CALL-WITH-FACET*) with :DIRECTION :IO or :OUTPUT does not call CHECK-NO-WATCHERS. This knob is intended to be bound locally for debugging purposes.

[function] CHECK-NO-WRITERS CUBE FACET-NAME MESSAGE-FORMAT &REST MESSAGE-ARGS

Signal an error if CUBE has facets (with names other than FACET-NAME) being written (i.e. direction is :IO or :OUTPUT).

[function] CHECK-NO-WATCHERS CUBE FACET-NAME MESSAGE-FORMAT &REST MESSAGE-ARGS

Signal an error if CUBE has facets (with names other than FACET-NAME) being regardless of the direction.

8 Lifetime

Lifetime management of facets is manual (but facets of garbage cubes are freed automatically by a finalizer, see MAKE-FACET*). One may destroy a single facet or all facets of a cube with DESTROY-FACET and DESTROY-CUBE, respectively. Also see Facet Barriers.

[function] DESTROY-FACET CUBE FACET-NAME

Free resources associated with the facet with FACET-NAME and remove it from FACETS(0 1) of CUBE.

[function] DESTROY-CUBE CUBE

Destroy all facets of CUBE with DESTROY-FACET.

In some cases it is useful to declare the intent to use a facet in the future to prevent its destruction. Hence, every facet has reference count which starts from 0. The reference count is incremented and decremented by ADD-FACET-REFERENCE-BY-NAME and REMOVE-FACET-REFERENCE-BY-NAME, respectively. If it is positive, then the facet will not be destroyed by explicit DESTROY-FACET and DESTROY-CUBE calls, but it will still be destroyed by the finalizer to prevent resource leaks caused by stray references.

[function] ADD-FACET-REFERENCE-BY-NAME CUBE FACET-NAME

Make sure FACET-NAME exists on CUBE and increment its reference count. Return the FACET behind FACET-NAME.

[function] REMOVE-FACET-REFERENCE-BY-NAME CUBE FACET-NAME

Decrement the reference count of the facet with FACET-NAME of CUBE. It is an error if the facet does not exists or if the reference count becomes negative.

[function] REMOVE-FACET-REFERENCE FACET

Decrement the reference count of FACET. It is an error if the facet is already destroyed or if the reference count becomes negative. This function has the same purpose as REMOVE-FACET-REFERENCE-BY-NAME, but by having a single FACET argument, it's more suited for use in finalizers because it does not keep the whole CUBE alive.

8.1 Facet Barriers

A facility to control lifetime of facets tied to a dynamic extent. Also see Lifetime.

[macro] WITH-FACET-BARRIER (CUBE-TYPE ENSURES DESTROYS) &BODY BODY

When BODY exits, destroy facets which:
- are of cubes with CUBE-TYPE
- have a facet name among DESTROYS
- were created in the dynamic extent of BODY
Before destroying the facets, it is ensured that facets with names among ENSURES are up-to-date. WITH-FACET-BARRIERs can be nested, in case of multiple barriers matching the cube's type and the created facet's name, the innermost one takes precedence.

The purpose of this macro is twofold. First, it makes it easy to temporarily work with a certain facet of many cubes without leaving newly created facets around. Second, it can be used to make sure that facets whose extent is tied to some dynamic boundary (such as the thread in which they were created) are destroyed.

[function] COUNT-BARRED-FACETS FACET-NAME &KEY (TYPE 'CUBE)

Count facets with FACET-NAME of cubes of TYPE which will be destroyed by a facet barrier.

Awesome

MAT Manual

Table of Contents

[in package MGL-MAT]

1 The MGL-MAT ASDF System

2 Links

3 Introduction

3.1 What's MGL-MAT?

3.2 What kind of matrices are supported?

3.3 Where to Get it?

4 Tutorial

5 Basics

6 Element types

7 Printing

8 Shaping

8.1 Comparison to Lisp Arrays

8.2 Functional Shaping

8.3 Destructive Shaping

9 Assembling

10 Caching

11 BLAS Operations

12 Destructive API

13 Non-destructive API

14 Mappings

15 Random numbers

16 I/O

17 Debugging

18 Facet API

18.1 Facets

18.2 Foreign arrays

18.3 CUDA

18.3.1 CUDA Memory Management

19 Writing Extensions

19.1 Lisp Extensions

19.2 CUDA Extensions

19.2.1 CUBLAS

19.2.2 CURAND

Cube Manual

Table of Contents

[in package MGL-CUBE]

1 Links

2 Introduction

3 Basics

4 Synchronization

5 Facets

6 Facet Extension API

7 The Default Implementation of CALL-WITH-FACET*

8 Lifetime

8.1 Facet Barriers

[generated by MGL-PAX]

1 The `MGL-MAT` ASDF System

7 The Default Implementation of `CALL-WITH-FACET*`