Mathematical Notation for LAPACK
Routines
Descriptions of LAPACK routines use the following
notation:
- AH
- For anM-by-NmatrixA, denotes the conjugate transposedN-by-Mmatrix with elements:For a real-valued matrix,.AH=AT
- x·y
- The dot product of two vectors, defined as:
- Ax=b
- A system of linear equations with ann-by-nmatrix, a right-hand side vectorA= {a}ij, and an unknown vectorb= {b}i.x= {x}i
- AX=B
- A set of systems with a common matrixAand multiple right-hand sides. The columns ofBare individual right-hand sides, and the columns ofXare the corresponding solutions.
- |x|
- the vector with elements|(absolute values ofx|i).xi
- |A|
- the matrix with elements|(absolute values ofa|ij).aij
- ||x||= max∞|ix|i
- The infinity-norm of the vectorx.
- ||A||= max∞iΣ|ja|ij
- The infinity-norm of the matrixA.
- ||A||= max1jΣ|ia|ij
- ||x||2
- The 2-norm of the vectorx:||(see the definition for Euclidean norm in thisx||2= (Σ|ix|i2)1/2= ||x||Etopic).
- ||A||2
- ||A||E
- κ(A) = ||A||·||A-1||
- The condition number of the matrixA.
- λi
- σi
- Singular values of the matrixA. They are equal to square roots of the eigenvalues ofAHA. (For more information, see Singular Value Decomposition).