Dense Matrix Functions
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For a detailed description of MPI Capability see Dense Matrix Parallelism Using MPI. |
Several decompositions and functions required for numerical linear algebra follow. The convention of enclosing optional quantities in brackets, “[ ]” is used.
Defined Array Functions | Matrix Operation |
|---|---|
S=SVD(A [,U=U, V=V]) | A = USVT |
E=EIG(A [[,B=B, D=D], V=V, W=W]) | (AV = VE), AVD = BVE (AW = WE), AWD = BWE |
R=CHOL(A) | A = RT R |
Q=ORTH(A [,R=R]) | (A = QR), QTQ = I |
U=UNIT(A) | [u1, …] = [a1/∥a1∥, …] |
F=DET(A) | Det(A) = determinant |
K=RANK(A) | rank(A) = rank |
P=NORM(A[,[type=]i]) | |
C=COND(A) | s1 / srank(A) |
Z=EYE(N) | Z = IN |
A=DIAG(X) | A = diag(x1, …) |
X=DIAGONALS(A) | x = (a11, …) |
Y=FFT (X,[WORK=W]); X=IFFT(Y,[WORK=W]) | Discrete Fourier Transform, Inverse |
Y=FFT_BOX (X,[WORK=W]); X=IFFT_BOX(Y,[WORK=W]) | Discrete Fourier Transform for Boxes, Inverse |
A=RAND(A) | Random numbers, 0 < A < 1 |
L=isNaN(A) | Test for NaN, if (l) then… |
In certain functions, the optional arguments are inputs while other optional arguments are outputs. To illustrate the example of the box SVD function, a code is given that computes the singular value decomposition and the reconstruction of the random matrix box, A, using the computed factors, R = USVT. Mathematically R = A, but this will be true, only approximately, due to rounding errors. The value units_of_error = ∥A ‑ R∥/(∥A∥ɛ), shows the merit of this approximation.
