Developer Reference

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  • 09/09/2020
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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

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Notice revision #20110804