<s>
LAPACK	B-Application
(	O
"	O
Linear	B-Application
Algebra	I-Application
Package	I-Application
"	O
)	O
is	O
a	O
standard	O
software	B-Library
library	I-Library
for	O
numerical	O
linear	O
algebra	O
.	O
</s>
<s>
It	O
provides	O
routines	O
for	O
solving	O
systems	O
of	O
linear	O
equations	O
and	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
eigenvalue	O
problems	O
,	O
and	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
It	O
also	O
includes	O
routines	O
to	O
implement	O
the	O
associated	O
matrix	B-Application
factorizations	I-Application
such	O
as	O
LU	O
,	O
QR	O
,	O
Cholesky	O
and	O
Schur	O
decomposition	O
.	O
</s>
<s>
LAPACK	B-Application
was	O
originally	O
written	O
in	O
FORTRAN	B-Application
77	O
,	O
but	O
moved	O
to	O
Fortran	B-Application
90	O
in	O
version	O
3.2	O
(	O
2008	O
)	O
.	O
</s>
<s>
LAPACK	B-Application
relies	O
on	O
an	O
underlying	O
BLAS	B-Application
implementation	O
to	O
provide	O
efficient	O
and	O
portable	O
computational	O
building	O
blocks	O
for	O
its	O
routines	O
.	O
</s>
<s>
LAPACK	B-Application
was	O
designed	O
as	O
the	O
successor	O
to	O
the	O
linear	O
equations	O
and	O
linear	O
least-squares	O
routines	O
of	O
LINPACK	B-Application
and	O
the	O
eigenvalue	O
routines	O
of	O
EISPACK	B-Application
.	O
</s>
<s>
LINPACK	B-Application
,	O
written	O
in	O
the	O
1970s	O
and	O
1980s	O
,	O
was	O
designed	O
to	O
run	O
on	O
the	O
then-modern	O
vector	B-Operating_System
computers	I-Operating_System
with	O
shared	O
memory	O
.	O
</s>
<s>
LAPACK	B-Application
,	O
in	O
contrast	O
,	O
was	O
designed	O
to	O
effectively	O
exploit	O
the	O
caches	O
on	O
modern	O
cache-based	O
architectures	O
and	O
the	O
instruction-level	B-Operating_System
parallelism	I-Operating_System
of	O
modern	O
superscalar	B-General_Concept
processors	I-General_Concept
,	O
and	O
thus	O
can	O
run	O
orders	O
of	O
magnitude	O
faster	O
than	O
LINPACK	B-Application
on	O
such	O
machines	O
,	O
given	O
a	O
well-tuned	O
BLAS	B-Application
implementation	O
.	O
</s>
<s>
LAPACK	B-Application
has	O
also	O
been	O
extended	O
to	O
run	O
on	O
distributed	B-Operating_System
memory	I-Operating_System
systems	O
in	O
later	O
packages	O
such	O
as	O
ScaLAPACK	B-Application
and	O
PLAPACK	B-Application
.	O
</s>
<s>
Netlib	O
LAPACK	B-Application
is	O
licensed	O
under	O
a	O
three-clause	O
BSD	B-Operating_System
style	I-Operating_System
license	O
,	O
a	O
permissive	B-License
free	I-License
software	I-License
license	I-License
with	O
few	O
restrictions	O
.	O
</s>
<s>
Subroutines	O
in	O
LAPACK	B-Application
have	O
a	O
naming	O
convention	O
which	O
makes	O
the	O
identifiers	O
very	O
compact	O
.	O
</s>
<s>
This	O
was	O
necessary	O
as	O
the	O
first	O
Fortran	B-Application
standards	O
only	O
supported	O
identifiers	O
up	O
to	O
six	O
characters	O
long	O
,	O
so	O
the	O
names	O
had	O
to	O
be	O
shortened	O
to	O
fit	O
into	O
this	O
limit	O
.	O
</s>
<s>
A	O
LAPACK	B-Application
subroutine	O
name	O
is	O
in	O
the	O
form	O
pmmaaa	O
,	O
where	O
:	O
</s>
<s>
S	O
,	O
D	O
stand	O
for	O
real	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
respectively	O
in	O
single	O
and	O
double	O
precision	O
,	O
while	O
C	B-Language
and	O
Z	O
stand	O
for	O
complex	O
arithmetic	O
with	O
respectively	O
single	O
and	O
double	O
precision	O
.	O
</s>
<s>
mm	O
is	O
a	O
two-letter	O
code	O
denoting	O
the	O
kind	O
of	O
matrix	B-Architecture
expected	O
by	O
the	O
algorithm	O
.	O
</s>
<s>
The	O
codes	O
for	O
the	O
different	O
kind	O
of	O
matrices	O
are	O
reported	O
below	O
;	O
the	O
actual	O
data	O
are	O
stored	O
in	O
a	O
different	O
format	O
depending	O
on	O
the	O
specific	O
kind	O
;	O
e.g.	O
,	O
when	O
the	O
code	O
DI	O
is	O
given	O
,	O
the	O
subroutine	O
expects	O
a	O
vector	O
of	O
length	O
n	O
containing	O
the	O
elements	O
on	O
the	O
diagonal	O
,	O
while	O
when	O
the	O
code	O
GE	O
is	O
given	O
,	O
the	O
subroutine	O
expects	O
an	O
array	O
containing	O
the	O
entries	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
SV	O
denotes	O
a	O
subroutine	O
to	O
solve	O
linear	O
system	O
,	O
while	O
R	B-Language
denotes	O
a	O
rank-1	O
update	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
subroutine	O
to	O
solve	O
a	O
linear	O
system	O
with	O
a	O
general	O
(	O
non-structured	O
)	O
matrix	B-Architecture
using	O
real	O
double-precision	O
arithmetic	O
is	O
called	O
DGESV	O
.	O
</s>
<s>
+Matrix	O
types	O
in	O
the	O
LAPACK	B-Application
naming	O
schemeNameDescriptionBDbidiagonal	O
matrixDIdiagonal	O
matrixGBgeneral	O
band	O
matrixGEgeneral	O
matrix	B-Architecture
(	O
i.e.	O
,	O
unsymmetric	B-Algorithm
,	O
in	O
some	O
cases	O
rectangular	O
)	O
GGgeneral	O
matrices	O
,	O
generalized	O
problem	O
(	O
i.e.	O
,	O
a	O
pair	O
of	O
general	O
matrices	O
)	O
GTgeneral	O
tridiagonal	B-Algorithm
matrixHB(complex )	O
Hermitian	B-Algorithm
band	O
matrixHE(complex )	O
Hermitian	B-Algorithm
matrixHGupper	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
,	O
generalized	O
problem	O
(	O
i.e.	O
</s>
<s>
Many	O
programming	O
environments	O
today	O
support	O
the	O
use	O
of	O
libraries	O
with	O
C	B-Language
binding	O
,	O
allowing	O
LAPACK	B-Application
routines	O
to	O
be	O
used	O
directly	O
so	O
long	O
as	O
a	O
few	O
restrictions	O
are	O
observed	O
.	O
</s>
<s>
Additionally	O
,	O
many	O
other	O
software	B-Library
libraries	I-Library
and	O
tools	O
for	O
scientific	O
and	O
numerical	O
computing	O
are	O
built	O
on	O
top	O
of	O
LAPACK	B-Application
,	O
such	O
as	O
R	B-Language
,	O
MATLAB	B-Language
,	O
and	O
SciPy	B-Application
.	O
</s>
<s>
Several	O
alternative	O
language	B-Application
bindings	I-Application
are	O
also	O
available	O
:	O
</s>
<s>
As	O
with	O
BLAS	B-Application
,	O
LAPACK	B-Application
is	O
sometimes	O
forked	O
or	O
rewritten	O
to	O
provide	O
better	O
performance	O
on	O
specific	O
systems	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>
Netlib	O
LAPACK	B-Application
The	O
official	O
LAPACK	B-Application
.	O
</s>
<s>
Netlib	O
ScaLAPACK	B-Application
Scalable	O
(	O
multicore	O
)	O
LAPACK	B-Application
,	O
built	O
on	O
top	O
of	O
PBLAS	B-Operating_System
.	O
</s>
<s>
Intel	B-Language
MKL	I-Language
Intel	O
's	O
Math	O
routines	O
for	O
their	O
x86	O
CPUs	O
.	O
</s>
<s>
OpenBLAS	B-Application
Open-source	O
reimplementation	O
of	O
BLAS	B-Application
and	O
LAPACK	B-Application
.	O
</s>
<s>
Gonum	O
LAPACK	B-Application
A	O
partial	O
native	O
Go	B-Application
implementation	O
.	O
</s>
<s>
Since	O
LAPACK	B-Application
typically	O
calls	O
underlying	O
BLAS	B-Application
routines	O
to	O
perform	O
the	O
bulk	O
of	O
its	O
computations	O
,	O
simply	O
linking	O
to	O
a	O
better-tuned	O
BLAS	B-Application
implementation	O
can	O
be	O
enough	O
to	O
significantly	O
improve	O
performance	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
LAPACK	B-Application
is	O
not	O
reimplemented	O
as	O
often	O
as	O
BLAS	B-Application
is	O
.	O
</s>
<s>
These	O
projects	O
provide	O
a	O
similar	O
functionality	O
to	O
LAPACK	B-Application
,	O
but	O
with	O
a	O
main	O
interface	O
differing	O
from	O
that	O
of	O
LAPACK	B-Application
:	O
</s>
<s>
Has	O
a	O
LAPACK-compatible	O
wrapper	O
.	O
</s>
<s>
Can	O
be	O
used	O
with	O
any	O
BLAS	B-Application
,	O
although	O
BLIS	B-Application
is	O
the	O
preferred	O
implementation	O
.	O
</s>
<s>
Eigen	B-Language
A	O
header	O
library	O
for	O
linear	O
algebra	O
.	O
</s>
<s>
Has	O
a	O
BLAS	B-Application
and	O
a	O
partial	O
LAPACK	B-Application
implementation	O
for	O
compatibility	O
.	O
</s>
<s>
MAGMA	O
Matrix	B-Architecture
Algebra	O
on	O
GPU	O
and	O
Multicore	O
Architectures	O
(	O
MAGMA	O
)	O
project	O
develops	O
a	O
dense	O
linear	O
algebra	O
library	O
similar	O
to	O
LAPACK	B-Application
but	O
for	O
heterogeneous	O
and	O
hybrid	O
architectures	O
including	O
multicore	O
systems	O
accelerated	O
with	O
GPGPUs	B-Architecture
.	O
</s>
<s>
PLASMA	O
The	O
Parallel	O
Linear	O
Algebra	O
for	O
Scalable	O
Multi-core	O
Architectures	O
(	O
PLASMA	O
)	O
project	O
is	O
a	O
modern	O
replacement	O
of	O
LAPACK	B-Application
for	O
multi-core	O
architectures	O
.	O
</s>
