Developer Reference

Contents

Matrix Layout for LAPACK Routines

There are two general methods of storing a two dimensional matrix in linear (one dimensional) memory: column-wise (column major order) or row-wise (row major order). Consider an
M
-by-
N
matrix
A
:

Column Major Layout

In column major layout the first index,
i
, of matrix elements
a
i
,
j
changes faster than the second index when accessing sequential memory locations. In other words, for 1
i
<
M
, if the element
a
i
,
j
is stored in a specific location in memory, the element
a
i
+1,
j
is stored in the next location, and, for 1
j
<
N
, the element
a
M
,
j
is stored in the location previous to element
a
1
,
j
+1
. So the matrix elements are located in memory according to this sequence:
{
a
1,1
a
2,1
...
a
M
,1
a
1,2
a
2,2
...
a
M
,2
... ...
a
1,
N
a
2,
N
...
a
M
,
N
}

Row Major Layout

In row major layout the second index,
j
, of matrix elements
a
i
,
j
changes faster than the first index when accessing sequential memory locations. In other words, for 1
j
<
N
, if the element
a
i
,
j
is stored in a specific location in memory, the element
a
i
,
j
+1
is stored in the next location, and, for 1
i
<
M
, the element
a
i
,
N
is stored in the location previous to element
a
i
+1,1
. So the matrix elements are located in memory according to this sequence:
{
a
1,1
a
1,2
...
a
1,
N
a
2,1
a
2,2
...
a
2,
N
... ...
a
N
,1
a
N
,2
...
a
M
,
N
}

Leading Dimension Parameter

A leading dimension parameter allows use of LAPACK routines on a submatrix of a larger matrix. For example, the submatrix
B
can be extracted from the original matrix
A
defined previously:
B
is formed from rows with indices
i
0
+ 1
to
i
0
+
K
and columns
j
0
+ 1
to
j
0
+
L
of matrix
A
. To specify matrix
B
, LAPACK routines require four parameters:
  • the number of rows
    K
    ;
  • the number of columns
    L
    ;
  • a pointer to the start of the array containing elements of
    B
    ;
  • the leading dimension of the array containing elements of
    B
    .
The leading dimension depends on the layout of the matrix:
  • Column major layout
    Leading dimension
    ldb
    =
    M
    , the number of rows of matrix
    A
    .
    Starting address: offset by
    i
    0
    +
    j
    0
    *
    ldb
    from
    a
    1,1
    .
  • Row major layout
    Leading dimension
    ldb
    =
    N
    , the number of columns of matrix
    A
    .
    Starting address: offset by
    i
    0
    *
    ldb
    +
    j
    0
    from
    a
    1,1
    .

Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804