Mathematical Notation

Descriptions of LAPACK routines use the following notation:

Ax = b

A system of linear equations with an n-by-n matrix A = {aij}, a right-hand side vector b = {bi}, and an unknown vector x = {xi}.

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 |xi| (absolute values of xi).

|A|

the matrix with elements |aij| (absolute values of aij).

||x|| = maxi|xi|

The infinity-norm of the vector x.

||A|| = maxiΣj|aij|

The infinity-norm of the matrix A.

||A||1 = maxjΣi|aij|

The one-norm of the matrix A. ||A||1 = ||AT|| = ||AH||

||x||2

The 2-norm of the vector x: ||x||2 = (Σi|xi|2)1/2 = ||x||E .

||A||2

The 2-norm (or spectral norm) of the matrix A.

||A||2 = maxiσi, ||A||22= max|x|=1(Ax·Ax).

||A||E

The Euclidean norm of the matrix A: ||A||E2 = ΣiΣj|aij|2 (for vectors, the Euclidean norm and the 2-norm are equal: ||x||E = ||x||2).

κ(A) = ||A|| ||A-1||

The condition number of the matrix A.

λi

Eigenvalues of the matrix A (for the definition of eigenvalues, see Eigenvalue Problems).

σi

Singular values of the matrix A. They are equal to square roots of the eigenvalues of AHA. (For more information, see Singular Value Decomposition).

q(x, y)

The acute angle between vectors x and y:

cos q(x, y) = |x·y| / (||x||2||y||2).

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