Contents

# Matrix Storage

The oneMKL BLAS and LAPACK routines for DPC++ use several matrix and vector storage formats. These are the same formats used in traditional Fortran BLAS/LAPACK.

## General Matrix

A general matrix
A
of
m
rows and
n
lda
is represented as a one dimensional array
a
of size of at least
lda
*
n
if column major layout is used and at least
lda
*
m
if row major layout is used. Before entry in any BLAS function using a general matrix, the leading
m
by
n
part of the array
a
must contain the matrix
A
. For column (respectively, row) major layout, the elements of each column (respectively, row) are contiguous in memory while the elements of each row (respectively, column) are at distance
lda
from the same elements in the previous row (respectively, column).
Visually, the matrix
is stored in memory as an array
For column major layout
For row major layout

## Triangular Matrix

A triangular matrix
A
of
n
rows and
n
lda
is represented as a one dimensional array
a
, of a size of at least
lda
*
n
. When column (respectively, row) major layout is used, the elements of each column (respectively, row) are contiguous in memory while the elements of each row (respectively, column) are at distance
lda
from the same elements in the previous row (respectively, column).
Before entry in any BLAS function using a triangular matrix,
• If
upper_lower = uplo::upper
n
by
n
upper triangular part of the array
a
must contain the upper triangular part of the matrix
A
. The strictly lower triangular part of the array
a
is not referenced. In other words, the matrix
is stored in memory as the array
• For column major layout
• For row major layout
• If
upper_lower = uplo::lower
n
by
n
lower triangular part of the array
a
must contain the lower triangular part of the matrix
A
. The strictly upper triangular part of the array
a
is not referenced. That is, the matrix
is stored in memory as the array
• For column major layout
• For row major layout

## Band Matrix

A general band matrix
A
of
m
rows and
n
columns with
kl
sub-diagonals,
ku
lda
is represented as a one dimensional array
a
of size at least
lda
*
n
(respectively,
lda
*
m
) if column (respectively, row) major layout is used.
Before entry in any BLAS function using a general band matrix, the leading (
kl
+
ku
+ 1
)
by
n
(respectively,
m
) part of the array
a
must contain the matrix
A
. This matrix must be supplied column-by-column (respectively, row-by-row), with the main diagonal of the matrix in row
ku
(respectively, column
kl
) of the array (0-based indexing), the first super-diagonal starting at position 1 (respectively, 0) in row (
ku
– 1) (respectively, column (
kl
+ 1)), the first sub-diagonal starting at position 0 (respectively, 1) in row (
ku
+ 1) (respectively, column (
kl
– 1)), and so on. Elements in the array
a
that do not correspond to elements in the band matrix (such as the top left
ku
-by-
ku
triangle) are not referenced.
Visually, the matrix
A
is stored in memory as an array
For column major layout
For row major layout
The following program segment transfers a band matrix from conventional full matrix storage (variable
matrix
ldm
) to band storage (variable
a
lda
):
• Using column major layout
``````for (j = 0; j < n; j++) {
k = ku – j;
for (i = max(0, j – ku); i < min(m, j + kl + 1); i++) {
a[(k + i) + j * lda] = matrix[i + j * ldm];
}
}``````
• Using row major layout
``````for (i = 0; i < m; i++) {
k = kl – i;
for (j = max(0, i – kl); j < min(n, i + ku + 1); j++) {
a[(k + j) + i * lda] = matrix[j + i * ldm];
}
}``````

## Triangular Band Matrix

A triangular band matrix
A
of
n
rows and
n
columns with
k
lda
is represented as a one dimensional array
a
of size at least
lda
*
n
.
Before entry in any BLAS function using a triangular band matrix,
• If
upper_lower = uplo::upper
k
+ 1) by
n
part of the array
a
must contain the upper triangular band part of the matrix
A
. When using column major layout, this matrix must be supplied column-by-column (respectively, row-by-row) with the main diagonal of the matrix in row (
k
) (respectively, column 0) of the array, the first super-diagonal starting at position 1 (respectively, 0) in row (
k
- 1) (respectively, column 1), and so on. Elements in the array
a
that do not correspond to elements in the triangular band matrix (such as the bottom left
k
by
k
triangle) are not referenced.
Visually, the matrix
is stored as an array
• For column major layout
• For row major layout
The following program segment transfers a band matrix from conventional full matrix storage (variable
matrix
ldm
) to band storage (variable
a
lda
):
• Using column major layout
``````for (j = 0; j < n; j++) {
m = k – j;
for (i = max(0, j – k); i <= j; i++) {
a[(m + i) + j * lda] = matrix[i + j * ldm];
}
}``````
• Using row major layout
``````for (i = 0; i < n; i++) {
m = –i;
for (j = i; j < min(n, i + k + 1); j++) {
a[(m + j) + i * lda] = matrix[j + i * ldm];
}
}``````
• If
upper_lower = uplo::lower
k
+ 1) by
n
part of the array
a
must contain the upper triangular band part of the matrix
A
. This matrix must be supplied column-by-column with the main diagonal of the matrix in row 0 of the array, the first sub-diagonal starting at position 0 in row 1, and so on. Elements in the array
a
that do not correspond to elements in the triangular band matrix (such as the bottom right
k
by
k
triangle) are not referenced.
That is, the matrix
is stored as the array
• For column major layout
• For row major layout
The following program segment transfers a band matrix from conventional full matrix storage (variable
matrix
ldm
) to band storage (variable
a
lda
):
• Using column major layout
``````for (j = 0; j < n; j++) {
m = –j;
for (i = j; i < min(n, j + k + 1); i++) {
a[(m + i) + j * lda] = matrix[i + j * ldm];
}
}``````
• Using row major layout
``````for (i = 0; i < n; i++) {
m = k – i;
for (j = max(0, i – k); j <= i; j++) {
a[(m + j) + i * lda] = matrix[j + i * ldm];
}
}``````

## Packed Triangular Matrix

A triangular matrix
A
of
n
rows and
n
columns is represented in packed format as a one dimensional array
a
of size at least (
n
*(
n
+ 1))/2. All elements in the upper or lower part of the matrix
A
are stored contiguously in the array
a
.
Before entry in any BLAS function using a triangular packed matrix,
• If
upper_lower = uplo::upper
, the first (
n
*(
n
+ 1))/2 elements in the array
a
must contain the upper triangular part of the matrix
A
packed sequentially, column by column so that
a
[0] contains
A
11
,
a
[1] and
a
[2] contain
A
12
and
A
22
respectively, and so on. Hence, the matrix
is stored as the array
• For column major layout
• For row major layout
• If
upper_lower = uplo::lower
, if column (respectively, row) major layout is used, the first (
n
*(
n
+ 1))/2 elements in the array
a
must contain the lower triangular part of the matrix
A
packed sequentially, column by column (respectively, row by row) so that
a
[0] contains
A
11
,
a
[1] and
a
[2] contain
A
21
and
A
31
respectively, and so on. The matrix
is stored as the array
• For column major layout
• For row major layout

## Vector

A vector
X
of
n
elements with increment
incx
is represented as a one dimensional array
x
of size at least (1 + (
n
- 1) * abs(
incx
)).
Visually, the vector
is stored in memory as an array

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