<s>
Basic	B-Application
Linear	I-Application
Algebra	I-Application
Subprograms	I-Application
(	O
BLAS	B-Application
)	O
is	O
a	O
specification	O
that	O
prescribes	O
a	O
set	O
of	O
low-level	O
routines	O
for	O
performing	O
common	O
linear	B-Language
algebra	I-Language
operations	O
such	O
as	O
vector	O
addition	O
,	O
scalar	O
multiplication	O
,	O
dot	O
products	O
,	O
linear	O
combinations	O
,	O
and	O
matrix	O
multiplication	O
.	O
</s>
<s>
They	O
are	O
the	O
de	O
facto	O
standard	O
low-level	O
routines	O
for	O
linear	B-Language
algebra	I-Language
libraries	O
;	O
the	O
routines	O
have	O
bindings	O
for	O
both	O
C	B-Language
(	O
"	O
CBLAS	B-Application
interface	O
"	O
)	O
and	O
Fortran	B-Application
(	O
"	O
BLAS	B-Application
interface	O
"	O
)	O
.	O
</s>
<s>
Although	O
the	O
BLAS	B-Application
specification	O
is	O
general	O
,	O
BLAS	B-Application
implementations	O
are	O
often	O
optimized	O
for	O
speed	O
on	O
a	O
particular	O
machine	O
,	O
so	O
using	O
them	O
can	O
bring	O
substantial	O
performance	O
benefits	O
.	O
</s>
<s>
BLAS	B-Application
implementations	O
will	O
take	O
advantage	O
of	O
special	O
floating	O
point	O
hardware	O
such	O
as	O
vector	O
registers	O
or	O
SIMD	B-Device
instructions	O
.	O
</s>
<s>
It	O
originated	O
as	O
a	O
Fortran	B-Application
library	B-Library
in	O
1979	O
and	O
its	O
interface	O
was	O
standardized	O
by	O
the	O
BLAS	B-Application
Technical	O
(	O
BLAST	O
)	O
Forum	O
,	O
whose	O
latest	O
BLAS	B-Application
report	O
can	O
be	O
found	O
on	O
the	O
netlib	B-Application
website	O
.	O
</s>
<s>
This	O
Fortran	B-Application
library	B-Library
is	O
known	O
as	O
the	O
reference	O
implementation	O
(	O
sometimes	O
confusingly	O
referred	O
to	O
as	O
the	O
BLAS	B-Application
library	B-Library
)	O
and	O
is	O
not	O
optimized	O
for	O
speed	O
but	O
is	O
in	O
the	O
public	O
domain	O
.	O
</s>
<s>
Most	O
computing	O
libraries	O
that	O
offer	O
linear	B-Language
algebra	I-Language
routines	O
conform	O
to	O
common	O
BLAS	B-Application
user	O
interface	O
command	O
structures	O
,	O
thus	O
queries	O
to	O
those	O
libraries	O
(	O
and	O
the	O
associated	O
results	O
)	O
are	O
often	O
portable	O
between	O
BLAS	B-Application
library	B-Library
branches	O
,	O
such	O
as	O
cuBLAS	O
(	O
nvidia	O
GPU	B-Architecture
,	O
GPGPU	B-Architecture
)	O
,	O
rocBLAS	O
(	O
amd	O
GPU	B-Architecture
,	O
GPGP	O
)	O
,	O
and	O
OpenBLAS	B-Application
.	O
</s>
<s>
Examples	O
of	O
CPU-based	O
BLAS	B-Application
library	B-Library
branches	O
include	O
:	O
OpenBLAS	B-Application
,	O
BLIS	B-Application
(	O
BLAS-like	O
Library	B-Library
Instantiation	O
Software	O
)	O
,	O
Arm	B-Architecture
Performance	O
Libraries	O
,	O
ATLAS	B-Language
,	O
and	O
Intel	B-Language
Math	I-Language
Kernel	I-Language
Library	I-Language
(	O
iMKL	B-Language
)	O
.	O
</s>
<s>
AMD	O
maintains	O
a	O
fork	O
of	O
BLIS	B-Application
that	O
is	O
optimized	O
for	O
the	O
AMD	O
platform	O
,	O
although	O
it	O
is	O
unclear	O
whether	O
integrated	O
ombudsmen	O
resources	O
are	O
present	O
in	O
that	O
particular	O
software-hardware	O
implementation	O
.	O
</s>
<s>
ATLAS	B-Language
is	O
a	O
portable	O
library	B-Library
that	O
automatically	O
optimizes	O
itself	O
for	O
an	O
arbitrary	O
architecture	O
.	O
</s>
<s>
iMKL	B-Language
is	O
a	O
freeware	O
and	O
proprietary	O
vendor	O
library	B-Library
optimized	O
for	O
x86	B-Operating_System
and	O
x86-64	B-Device
with	O
a	O
performance	O
emphasis	O
on	O
Intel	O
processors	O
.	O
</s>
<s>
OpenBLAS	B-Application
is	O
an	O
open-source	B-Application
library	B-Library
that	O
is	O
hand-optimized	O
for	O
many	O
of	O
the	O
popular	O
architectures	O
.	O
</s>
<s>
The	O
LINPACK	B-Device
benchmarks	I-Device
rely	O
heavily	O
on	O
the	O
BLAS	B-Application
routine	O
gemm	O
for	O
its	O
performance	O
measurements	O
.	O
</s>
<s>
Many	O
numerical	O
software	O
applications	O
use	O
BLAS-compatible	O
libraries	O
to	O
do	O
linear	B-Language
algebra	I-Language
computations	O
,	O
including	O
LAPACK	B-Application
,	O
LINPACK	B-Application
,	O
Armadillo	B-Language
,	O
GNU	B-Language
Octave	I-Language
,	O
Mathematica	B-Language
,	O
MATLAB	B-Language
,	O
NumPy	B-Application
,	O
R	B-Language
,	O
and	O
Julia	B-Application
.	O
</s>
<s>
With	O
the	O
advent	O
of	O
numerical	O
programming	O
,	O
sophisticated	O
subroutine	B-Library
libraries	I-Library
became	O
useful	O
.	O
</s>
<s>
The	O
language	O
of	O
choice	O
was	O
FORTRAN	B-Application
.	O
</s>
<s>
The	O
most	O
prominent	O
numerical	O
programming	B-Library
library	I-Library
was	O
IBM	O
's	O
Scientific	O
Subroutine	O
Package	O
(	O
SSP	O
)	O
.	O
</s>
<s>
These	O
subroutine	B-Library
libraries	I-Library
allowed	O
programmers	O
to	O
concentrate	O
on	O
their	O
specific	O
problems	O
and	O
avoid	O
re-implementing	O
well-known	O
algorithms	O
.	O
</s>
<s>
The	O
library	B-Library
routines	I-Library
would	O
also	O
be	O
better	O
than	O
average	O
implementations	O
;	O
matrix	O
algorithms	O
,	O
for	O
example	O
,	O
might	O
use	O
full	O
pivoting	O
to	O
get	O
better	O
numerical	O
accuracy	O
.	O
</s>
<s>
The	O
library	B-Library
routines	I-Library
would	O
also	O
have	O
more	O
efficient	O
routines	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
library	B-Library
may	O
include	O
a	O
program	O
to	O
solve	O
a	O
matrix	O
that	O
is	O
upper	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
programs	O
have	O
many	O
common	O
low-level	O
operations	O
(	O
the	O
so-called	O
"	O
kernel	B-Operating_System
"	O
operations	O
,	O
not	O
related	O
to	O
operating	O
systems	O
)	O
.	O
</s>
<s>
Between	O
1973	O
and	O
1977	O
,	O
several	O
of	O
these	O
kernel	B-Operating_System
operations	O
were	O
identified	O
.	O
</s>
<s>
These	O
kernel	B-Operating_System
operations	O
became	O
defined	O
subroutines	O
that	O
math	O
libraries	O
could	O
call	O
.	O
</s>
<s>
The	O
kernel	B-Operating_System
calls	O
had	O
advantages	O
over	O
hard-coded	O
loops	O
:	O
the	O
library	B-Library
routine	I-Library
would	O
be	O
more	O
readable	O
,	O
there	O
were	O
fewer	O
chances	O
for	O
bugs	O
,	O
and	O
the	O
kernel	B-Operating_System
implementation	O
could	O
be	O
optimized	O
for	O
speed	O
.	O
</s>
<s>
A	O
specification	O
for	O
these	O
kernel	B-Operating_System
operations	O
using	O
scalars	O
and	O
vectors	O
,	O
the	O
level-1	O
Basic	O
Linear	B-Language
Algebra	I-Language
Subroutines	O
(	O
BLAS	B-Application
)	O
,	O
was	O
published	O
in	O
1979	O
.	O
</s>
<s>
BLAS	B-Application
was	O
used	O
to	O
implement	O
the	O
linear	B-Language
algebra	I-Language
subroutine	B-Library
library	I-Library
LINPACK	B-Application
.	O
</s>
<s>
The	O
BLAS	B-Application
abstraction	O
allows	O
customization	O
for	O
high	O
performance	O
.	O
</s>
<s>
For	O
example	O
,	O
LINPACK	B-Application
is	O
a	O
general	O
purpose	O
library	B-Library
that	O
can	O
be	O
used	O
on	O
many	O
different	O
machines	O
without	O
modification	O
.	O
</s>
<s>
LINPACK	B-Application
could	O
use	O
a	O
generic	O
version	O
of	O
BLAS	B-Application
.	O
</s>
<s>
To	O
gain	O
performance	O
,	O
different	O
machines	O
might	O
use	O
tailored	O
versions	O
of	O
BLAS	B-Application
.	O
</s>
<s>
As	O
computer	O
architectures	O
became	O
more	O
sophisticated	O
,	O
vector	B-Operating_System
machines	I-Operating_System
appeared	O
.	O
</s>
<s>
BLAS	B-Application
for	O
a	O
vector	O
machine	O
could	O
use	O
the	O
machine	O
's	O
fast	O
vector	O
operations	O
.	O
</s>
<s>
(	O
While	O
vector	B-Operating_System
processors	I-Operating_System
eventually	O
fell	O
out	O
of	O
favor	O
,	O
vector	O
instructions	O
in	O
modern	O
CPUs	O
are	O
essential	O
for	O
optimal	O
performance	O
in	O
BLAS	B-Application
routines	O
.	O
)	O
</s>
<s>
Consequently	O
,	O
BLAS	B-Application
was	O
augmented	O
from	O
1984	O
to	O
1986	O
with	O
level-2	O
kernel	B-Operating_System
operations	O
that	O
concerned	O
vector-matrix	O
operations	O
.	O
</s>
<s>
Many	O
computers	O
have	O
cache	B-General_Concept
memory	I-General_Concept
that	O
is	O
much	O
faster	O
than	O
main	O
memory	O
;	O
keeping	O
matrix	O
manipulations	O
localized	O
allows	O
better	O
usage	O
of	O
the	O
cache	B-General_Concept
.	O
</s>
<s>
In	O
1987	O
and	O
1988	O
,	O
the	O
level	O
3	O
BLAS	B-Application
were	O
identified	O
to	O
do	O
matrix-matrix	O
operations	O
.	O
</s>
<s>
The	O
level	O
3	O
BLAS	B-Application
encouraged	O
block-partitioned	O
algorithms	O
.	O
</s>
<s>
The	O
LAPACK	B-Application
library	B-Library
uses	O
level	O
3	O
BLAS	B-Application
.	O
</s>
<s>
The	O
original	O
BLAS	B-Application
concerned	O
only	O
densely	O
stored	O
vectors	O
and	O
matrices	O
.	O
</s>
<s>
Further	O
extensions	O
to	O
BLAS	B-Application
,	O
such	O
as	O
for	O
sparse	B-Algorithm
matrices	I-Algorithm
,	O
have	O
been	O
addressed	O
.	O
</s>
<s>
BLAS	B-Application
functionality	O
is	O
categorized	O
into	O
three	O
sets	O
of	O
routines	O
called	O
"	O
levels	O
"	O
,	O
which	O
correspond	O
to	O
both	O
the	O
chronological	O
order	O
of	O
definition	O
and	O
publication	O
,	O
as	O
well	O
as	O
the	O
degree	O
of	O
the	O
polynomial	O
in	O
the	O
complexities	O
of	O
algorithms	O
;	O
Level	O
1	O
BLAS	B-Application
operations	O
typically	O
take	O
linear	O
time	O
,	O
,	O
Level	O
2	O
operations	O
quadratic	O
time	O
and	O
Level	O
3	O
operations	O
cubic	O
time	O
.	O
</s>
<s>
Modern	O
BLAS	B-Application
implementations	O
typically	O
provide	O
all	O
three	O
levels	O
.	O
</s>
<s>
Design	O
of	O
the	O
Level	O
2	O
BLAS	B-Application
started	O
in	O
1984	O
,	O
with	O
results	O
published	O
in	O
1988	O
.	O
</s>
<s>
The	O
Level	O
2	O
subroutines	O
are	O
especially	O
intended	O
to	O
improve	O
performance	O
of	O
programs	O
using	O
BLAS	B-Application
on	O
vector	B-Operating_System
processors	I-Operating_System
,	O
where	O
Level	O
1	O
BLAS	B-Application
are	O
suboptimal	O
"	O
because	O
they	O
hide	O
the	O
matrix-vector	O
nature	O
of	O
the	O
operations	O
from	O
the	O
compiler.	O
"	O
</s>
<s>
where	O
is	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
,	O
among	O
other	O
functionality	O
.	O
</s>
<s>
Due	O
to	O
the	O
ubiquity	O
of	O
matrix	O
multiplications	O
in	O
many	O
scientific	O
applications	O
,	O
including	O
for	O
the	O
implementation	O
of	O
the	O
rest	O
of	O
Level	O
3	O
BLAS	B-Application
,	O
and	O
because	O
faster	O
algorithms	O
exist	O
beyond	O
the	O
obvious	O
repetition	O
of	O
matrix-vector	O
multiplication	O
,	O
gemm	O
is	O
a	O
prime	O
target	O
of	O
optimization	O
for	O
BLAS	B-Application
implementers	O
.	O
</s>
<s>
E.g.	O
,	O
by	O
decomposing	O
one	O
or	O
both	O
of	O
,	O
into	O
block	B-Algorithm
matrices	I-Algorithm
,	O
gemm	O
can	O
be	O
implemented	O
recursively	O
.	O
</s>
<s>
This	O
decomposition	O
allows	O
for	O
better	O
locality	B-General_Concept
of	I-General_Concept
reference	I-General_Concept
both	O
in	O
space	O
and	O
time	O
of	O
the	O
data	O
used	O
in	O
the	O
product	O
.	O
</s>
<s>
This	O
,	O
in	O
turn	O
,	O
takes	O
advantage	O
of	O
the	O
cache	B-General_Concept
on	O
the	O
system	O
.	O
</s>
<s>
For	O
systems	O
with	O
more	O
than	O
one	O
level	O
of	O
cache	B-General_Concept
,	O
the	O
blocking	O
can	O
be	O
applied	O
a	O
second	O
time	O
to	O
the	O
order	O
in	O
which	O
the	O
blocks	O
are	O
used	O
in	O
the	O
computation	O
.	O
</s>
<s>
Both	O
of	O
these	O
levels	O
of	O
optimization	O
are	O
used	O
in	O
implementations	O
such	O
as	O
ATLAS	B-Language
.	O
</s>
<s>
More	O
recently	O
,	O
implementations	O
by	O
Kazushige	O
Goto	O
have	O
shown	O
that	O
blocking	O
only	O
for	O
the	O
L2	O
cache	B-General_Concept
,	O
combined	O
with	O
careful	O
amortizing	B-General_Concept
of	O
copying	O
to	O
contiguous	O
memory	O
to	O
reduce	O
TLB	B-Architecture
misses	O
,	O
is	O
superior	O
to	O
ATLAS	B-Language
.	O
</s>
<s>
A	O
highly	O
tuned	O
implementation	O
based	O
on	O
these	O
ideas	O
is	O
part	O
of	O
the	O
GotoBLAS	B-Application
,	O
OpenBLAS	B-Application
and	O
BLIS	B-Application
.	O
</s>
<s>
A	O
common	O
variation	O
of	O
is	O
the	O
,	O
which	O
calculates	O
a	O
complex	O
product	O
using	O
"	O
three	O
real	O
matrix	O
multiplications	O
and	O
five	O
real	O
matrix	O
additions	O
instead	O
of	O
the	O
conventional	O
four	O
real	O
matrix	O
multiplications	O
and	O
two	O
real	O
matrix	O
additions	O
"	O
,	O
an	O
algorithm	O
similar	O
to	O
Strassen	B-Algorithm
algorithm	I-Algorithm
first	O
described	O
by	O
Peter	O
Ungar	O
.	O
</s>
<s>
Accelerate	O
Apple	O
's	O
framework	O
for	O
macOS	B-Application
and	O
iOS	B-Operating_System
,	O
which	O
includes	O
tuned	O
versions	O
of	O
BLAS	B-Application
and	O
LAPACK	B-Application
.	O
</s>
<s>
Arm	B-Architecture
Performance	O
Libraries	O
Arm	B-Architecture
Performance	O
Libraries	O
,	O
supporting	O
Arm	B-Architecture
64-bit	O
AArch64-based	O
processors	O
,	O
available	O
from	O
Arm	B-Architecture
.	O
</s>
<s>
ATLAS	B-Language
Automatically	B-Language
Tuned	I-Language
Linear	I-Language
Algebra	I-Language
Software	I-Language
,	O
an	O
open	B-Application
source	I-Application
implementation	O
of	O
BLAS	B-Application
APIs	B-Application
for	O
C	B-Language
and	O
Fortran	B-Application
77	I-Application
.	O
</s>
<s>
BLIS	B-Application
BLAS-like	O
Library	B-Library
Instantiation	O
Software	O
framework	O
for	O
rapid	O
instantiation	O
.	O
</s>
<s>
BLIS	B-Application
is	O
a	O
complete	O
refactoring	O
of	O
the	O
GotoBLAS	B-Application
that	O
reduces	O
the	O
amount	O
of	O
code	O
that	O
must	O
be	O
written	O
for	O
a	O
given	O
platform	O
.	O
</s>
<s>
C++	B-Language
AMP	I-Language
BLAS	B-Application
The	O
C++	B-Language
AMP	I-Language
BLAS	B-Application
Library	B-Library
is	O
an	O
open	B-Application
source	I-Application
implementation	O
of	O
BLAS	B-Application
for	O
Microsoft	O
's	O
AMP	O
language	O
extension	O
for	O
Visual	O
C++	B-Language
.	O
</s>
<s>
cuBLAS	O
Optimized	O
BLAS	B-Application
for	O
NVIDIA	O
based	O
GPU	B-Architecture
cards	O
,	O
requiring	O
few	O
additional	O
library	B-Library
calls	O
.	O
</s>
<s>
NVBLAS	O
Optimized	O
BLAS	B-Application
for	O
NVIDIA	O
based	O
GPU	B-Architecture
cards	O
,	O
providing	O
only	O
Level	O
3	O
functions	O
,	O
but	O
as	O
direct	O
drop-in	O
replacement	O
for	O
other	O
BLAS	B-Application
libraries	O
.	O
</s>
<s>
clBLAS	O
An	O
OpenCL	B-Application
implementation	O
of	O
BLAS	B-Application
by	O
AMD	O
.	O
</s>
<s>
clBLAST	O
A	O
tuned	O
OpenCL	B-Application
implementation	O
of	O
most	O
of	O
the	O
BLAS	B-Application
api	B-Application
.	O
</s>
<s>
Eigen	B-Language
BLAS	B-Application
A	O
Fortran	B-Application
77	I-Application
and	O
C	B-Language
BLAS	B-Application
library	B-Library
implemented	O
on	O
top	O
of	O
the	O
MPL-licensed	O
Eigen	B-Language
library	I-Language
,	O
supporting	O
x86	B-Operating_System
,	O
x86-64	B-Device
,	O
ARM	B-Architecture
(	O
NEON	O
)	O
,	O
and	O
PowerPC	B-Architecture
architectures	O
.	O
</s>
<s>
ESSL	O
IBM	O
's	O
Engineering	O
and	O
Scientific	O
Subroutine	B-Library
Library	I-Library
,	O
supporting	O
the	O
PowerPC	B-Architecture
architecture	O
under	O
AIX	B-Application
and	O
Linux	B-Application
.	O
</s>
<s>
GotoBLAS	B-Application
Kazushige	O
Goto	O
's	O
BSD-licensed	O
implementation	O
of	O
BLAS	B-Application
,	O
tuned	O
in	O
particular	O
for	O
Intel	O
Nehalem/Atom	O
,	O
VIA	O
Nanoprocessor	B-Device
,	O
AMD	B-General_Concept
Opteron	I-General_Concept
.	O
</s>
<s>
GNU	B-Application
Scientific	I-Application
Library	I-Application
Multi-platform	B-Operating_System
implementation	O
of	O
many	O
numerical	O
routines	O
.	O
</s>
<s>
Contains	O
a	O
CBLAS	B-Application
interface	O
.	O
</s>
<s>
HP	O
MLIB	O
HP	O
's	O
Math	O
library	B-Library
supporting	O
IA-64	B-General_Concept
,	O
PA-RISC	B-Device
,	O
x86	B-Operating_System
and	O
Opteron	B-General_Concept
architecture	O
under	O
HP-UX	B-Application
and	O
Linux	B-Application
.	O
</s>
<s>
Intel	B-Language
MKL	I-Language
The	O
Intel	B-Language
Math	I-Language
Kernel	I-Language
Library	I-Language
,	O
supporting	O
x86	B-Operating_System
32-bits	O
and	O
64-bits	O
,	O
available	O
free	O
from	O
Intel	O
.	O
</s>
<s>
Includes	O
optimizations	O
for	O
Intel	B-Device
Pentium	I-Device
,	O
Core	B-Device
and	O
Intel	B-Device
Xeon	I-Device
CPUs	I-Device
and	O
Intel	B-General_Concept
Xeon	I-General_Concept
Phi	I-General_Concept
;	O
support	O
for	O
Linux	B-Application
,	O
Windows	B-Application
and	O
macOS	B-Application
.	O
</s>
<s>
Netlib	B-Application
BLAS	B-Application
The	O
official	O
reference	O
implementation	O
on	O
Netlib	B-Application
,	O
written	O
in	O
Fortran	B-Application
77	I-Application
.	O
</s>
<s>
Netlib	B-Application
CBLAS	B-Application
Reference	O
C	B-Language
interface	O
to	O
the	O
BLAS	B-Application
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
(	O
and	O
popular	O
)	O
to	O
call	O
the	O
Fortran	B-Application
BLAS	B-Application
from	O
C	B-Language
.	O
</s>
<s>
OpenBLAS	B-Application
Optimized	O
BLAS	B-Application
based	O
on	O
GotoBLAS	B-Application
,	O
supporting	O
x86	B-Operating_System
,	O
x86-64	B-Device
,	O
MIPS	B-Device
and	O
ARM	B-Architecture
processors	I-Architecture
.	O
</s>
<s>
PDLIB/SX	O
NEC	O
's	O
Public	O
Domain	O
Mathematical	O
Library	B-Library
for	O
the	O
NEC	B-Device
SX-4	I-Device
system	O
.	O
</s>
<s>
rocBLAS	O
Implementation	O
that	O
runs	O
on	O
AMD	O
GPUs	B-Architecture
via	O
ROCm	B-Operating_System
.	O
</s>
<s>
SGI	O
's	O
Scientific	O
Computing	O
Software	B-Library
Library	I-Library
contains	O
BLAS	B-Application
and	O
LAPACK	B-Application
implementations	O
for	O
SGI	O
's	O
Irix	B-Operating_System
workstations	O
.	O
</s>
<s>
Sun	O
Performance	O
Library	B-Library
Optimized	O
BLAS	B-Application
and	O
LAPACK	B-Application
for	O
SPARC	B-Architecture
,	O
Core	B-Device
and	O
AMD64	B-Device
architectures	O
under	O
Solaris	O
8	O
,	O
9	O
,	O
and	O
10	O
as	O
well	O
as	O
Linux	B-Application
.	O
</s>
<s>
uBLAS	B-Language
A	O
generic	O
C++	B-Language
template	B-Language
class	I-Language
library	B-Library
providing	O
BLAS	B-Application
functionality	O
.	O
</s>
<s>
Part	O
of	O
the	O
Boost	B-Language
library	I-Language
.	O
</s>
<s>
Moreover	O
,	O
uBLAS	B-Language
focuses	O
on	O
correctness	O
of	O
the	O
algorithms	O
using	O
advanced	O
C++	B-Language
features	O
.	O
</s>
<s>
Armadillo	B-Language
Armadillo	B-Language
is	O
a	O
C++	B-Language
linear	B-Language
algebra	I-Language
library	B-Library
aiming	O
towards	O
a	O
good	O
balance	O
between	O
speed	O
and	O
ease	O
of	O
use	O
.	O
</s>
<s>
It	O
employs	O
template	O
classes	O
,	O
and	O
has	O
optional	O
links	O
to	O
BLAS/ATLAS	O
and	O
LAPACK	B-Application
.	O
</s>
<s>
LAPACK	B-Application
LAPACK	B-Application
is	O
a	O
higher	O
level	O
Linear	B-Language
Algebra	I-Language
library	B-Library
built	O
upon	O
BLAS	B-Application
.	O
</s>
<s>
Like	O
BLAS	B-Application
,	O
a	O
reference	O
implementation	O
exists	O
,	O
but	O
many	O
alternatives	O
like	O
libFlame	O
and	O
MKL	O
exist	O
.	O
</s>
<s>
Mir	O
An	O
LLVM-accelerated	O
generic	O
numerical	O
library	B-Library
for	O
science	O
and	O
machine	O
learning	O
written	O
in	O
D	B-Application
.	O
It	O
provides	O
generic	O
linear	B-Language
algebra	I-Language
subprograms	O
(	O
GLAS	O
)	O
.	O
</s>
<s>
It	O
can	O
be	O
built	O
on	O
a	O
CBLAS	B-Application
implementation	O
.	O
</s>
<s>
Elemental	O
Elemental	O
is	O
an	O
open	B-Application
source	I-Application
software	I-Application
for	O
distributed-memory	B-Operating_System
dense	O
and	O
sparse-direct	O
linear	B-Language
algebra	I-Language
and	O
optimization	O
.	O
</s>
<s>
HASEM	O
is	O
a	O
C++	B-Language
template	O
library	B-Library
,	O
being	O
able	O
to	O
solve	O
linear	O
equations	O
and	O
to	O
compute	O
eigenvalues	O
.	O
</s>
<s>
LAMA	O
The	O
Library	B-Library
for	O
Accelerated	O
Math	O
Applications	O
(	O
LAMA	O
)	O
is	O
a	O
C++	B-Language
template	O
library	B-Library
for	O
writing	O
numerical	O
solvers	O
targeting	O
various	O
kinds	O
of	O
hardware	O
(	O
e.g.	O
</s>
<s>
MTL4	O
The	O
Matrix	B-Language
Template	I-Language
Library	I-Language
version	O
4	O
is	O
a	O
generic	O
C++	B-Language
template	O
library	B-Library
providing	O
sparse	O
and	O
dense	O
BLAS	B-Application
functionality	O
.	O
</s>
<s>
MTL4	O
establishes	O
an	O
intuitive	O
interface	O
(	O
similar	O
to	O
MATLAB	B-Language
)	O
and	O
broad	O
applicability	O
thanks	O
to	O
generic	B-Language
programming	I-Language
.	O
</s>
<s>
Several	O
extensions	O
to	O
BLAS	B-Application
for	O
handling	O
sparse	B-Algorithm
matrices	I-Algorithm
have	O
been	O
suggested	O
over	O
the	O
course	O
of	O
the	O
library	B-Library
's	O
history	O
;	O
a	O
small	O
set	O
of	O
sparse	B-Algorithm
matrix	I-Algorithm
kernel	B-Operating_System
routines	O
was	O
finally	O
standardized	O
in	O
2002	O
.	O
</s>
<s>
The	O
traditional	O
BLAS	B-Application
functions	O
have	O
been	O
also	O
ported	O
to	O
architectures	O
that	O
support	O
large	O
amounts	O
of	O
parallelism	O
such	O
as	O
GPUs	B-Architecture
.	O
</s>
<s>
Here	O
,	O
the	O
traditional	O
BLAS	B-Application
functions	O
provide	O
typically	O
good	O
performance	O
for	O
large	O
matrices	O
.	O
</s>
<s>
To	O
address	O
this	O
issue	O
,	O
in	O
2017	O
a	O
batched	O
version	O
of	O
the	O
BLAS	B-Application
function	O
has	O
been	O
specified	O
.	O
</s>
<s>
Batched	O
BLAS	B-Application
functions	O
can	O
be	O
a	O
versatile	O
tool	O
and	O
allow	O
e.g.	O
</s>
<s>
a	O
fast	O
implementation	O
of	O
exponential	B-Algorithm
integrators	I-Algorithm
and	O
Magnus	O
integrators	O
that	O
handle	O
long	O
integration	O
periods	O
with	O
many	O
time	O
steps	O
.	O
</s>
<s>
Here	O
,	O
the	O
matrix	O
exponentiation	O
,	O
the	O
computationally	O
expensive	O
part	O
of	O
the	O
integration	O
,	O
can	O
be	O
implemented	O
in	O
parallel	O
for	O
all	O
time-steps	O
by	O
using	O
Batched	O
BLAS	B-Application
functions	O
.	O
</s>
