Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

Mathematical Notation for LAPACK Routines

Descriptions of LAPACK routines use the following notation:
A
H
For an
M
-by-
N
 matrix
A
, denotes the conjugate transposed
N
-by-
M
 matrix with elements:
For a real-valued matrix,
A
H
 =
A
T
.
x
·
y
The dot product of two vectors, defined as:
Ax
 =
b
A system of linear equations with an
n
-by-
n
 matrix
A
 = {
a
i
j
}
, a right-hand side vector
b
 = {
b
i
}
, and an unknown vector
x
 = {
x
i
}
.
AX
 =
B
A set of systems with a common matrix
A
 and multiple right-hand sides. The columns of
B
 are individual right-hand sides, and the columns of
X
 are the corresponding solutions.
|
x
|
the vector with elements
|
x
i
|
 (absolute values of
x
i
).
|
A
|
the matrix with elements
|
a
i
j
|
 (absolute values of
a
i
j
).
||
x
||
∞
 = max
i
|
x
i
|
The infinity-norm of the vector
x
.
||
A
||
∞
 = max
i
Σ
j
|
a
ij
|
The infinity-norm of the matrix
A
.
||
A
||
1
 = max
j
Σ
i
|
a
ij
|
The one-norm of the matrix
A
.
||
A
||
1
 = ||
A
T
||
∞
 = ||
A
H
||
∞
||
x
||
2
The 2-norm of the vector
x
:
||
x
||
2
 = (
Σ
i
|
x
i
|
2
)
1/2
 = ||
x
||
E
 (see the definition for Euclidean norm in this
topic
).
||
A
||
2
The 2-norm (or spectral norm) of the matrix
A
.
||
A
||
E
The Euclidean norm of the matrix
A
:
||
A
||
E
2
 =
Σ
i
Σ
j
|
a
ij
|
2
.
κ
(
A
) = ||
A
||
·
||
A
-1
||
The condition number of the matrix
A
.
λ
i
Eigenvalues of the matrix
A
 (for the definition of eigenvalues, see Eigenvalue Problems).
σ
i
Singular values of the matrix
A
. They are equal to square roots of the eigenvalues of
A
H
A
. (For more information, see Singular Value Decomposition).

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.