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74 Polynomial Approximation of Differential Equations
(4.3.10) Φ1 =
(4.3.11) Φ2 =
(4.3.12) Φ3 =
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φ0 φ0 0 0 0 0 0 0
φ0 φ4 0 0 0 0 0 0
0 0 φ0 φ2 0 0 0 0
0 0 φ0 φ6 0 0 0 0
0 0 0 0 φ0 φ1 0 0
0 0 0 0 φ0 φ5 0 0
0 0 0 0 0 0 φ0 φ3 0 0 0 0 0 0 φ0 φ7 φ0 0 φ0 0 0 0 0 0
0 φ0 0 φ0 0 0 0 0
φ0 0 φ4 0 0 0 0 0
0 φ0 0 φ4 0 0 0 0
0 0 0 0 φ0 0 φ2 0
0 0 0 0 0 φ0 0 φ2 0 0 0 0 φ0 0 φ6 0
0 0 0 0 0 φ0 0 φ6 φ0 0 0 0 φ0 0 0 0
0 φ0 0 0 0 φ0 0 0
0 0 φ0 0 0 0 φ0 0
0 0 0 φ0 0 0 0 φ0 φ0 0 0 0 φ4 0 0 0
0 φ0 0 0 0 φ4 0 0
0 0 φ0 0 0 0 φ4 0
0 0 0 φ0 0 0 0 φ4 This splitting is the kernel of the FFT algorithm. In general, starting from the n×n full
matrix Φ (where n is a power of 2), we can find a suitable sequence of n × n matrices
Φk, 1 ≤ k ≤ log 2n, whose product Φ ∗ := log 2 n
k=1 Φk allows the recovery of Φ. As the
reader can observe, the data entries in the left-hand side of (4.3.9) have been permuted.
Therefore, Φ is obtained after a suitable recombination of the rows of Φ ∗ .
The new matrices are very sparse. Actually, a vector multiplication just involves
two nonzero entries for each row. The FFT algorithm takes advantage of the reg-
ular structure of the Φk’s. This reduces the matrix-vector multiplication for any
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