By Robert A. van de Geijn, Enrique S. Quintana-Ortí
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Extra info for The Science of Programming Matrix Computations
10 Consider 0 1 2 B 5 C C y=B @ 1 A, 8 1 3 x = @ 0 A, 1 0 and 0 3 B 1 A=B @ 9 7 1 2 0 1 The matrix-vector product y := Ax + y can be computed as several axpys: 222 0 1 0 3 666 B 1 C B B C 66 B y := 6 444 3 @ 9 A + 0 @ 7 as well as via repeated apdots: 0 B B B B B B B B B B B y := B B B B B B B B B B @ (3 1 (1 2 (9 0 (7 1 0 3 2) @ 0 1 0 3 2) @ 0 1 0 3 2) @ 0 1 0 3 0) @ 0 1 1 1 3 0 2 1 C B 2 7 2 C 7 B + 1@ 2 0 A 5 0 1 1 0 2 C 7 B 5 C C C 7+B A 5 @ 1 A 8 1 3 1 2 2 C C. 2 A 0 1 0 13 7 B 10 C C 7=B 5 @ 30 A , 29 3 1 A+2 C C C C C 1 C C A+5 C 0 C C B C B C=@ 1 C C A+1 C C C C C 1 C C A+8 A 1 0 13 3·3+1·0+2·1+2 B 1·3+2·0+2·1+5 C C = B 10 9 · 3 + 0 · 0 + 2 · 1 + 1 A @ 30 29 7·3+1·0+0·1+9 1 C C.
3 Let F : Rn → Rm be a function that maps a vector from Rn to a vector in Rm . Then F is said to be a linear transformation if F(αx + y) = αF(x) + F(y) for all α ∈ R and x, y ∈ Rn . 2. Linear Transformations and Matrices 21 Consider the unit basis vectors, ej ∈ Rn , 0 ≤ j < n, which are defined by the vectors of all zeroes except for the jth element, which equals 1: 0 .. . j zeroes (elements 0, 1, . . , j − 1) 0 element j ej = 1 ← 0 . n − j − 1 zeroes (elements j + 1, j + 2, .
Solving Triangular Linear Systems of Equations 35 Step 3: Choosing a Loop-guard. For either of the two loop-invariants, the loop-guard “G : m(bT ) < m(b)” has the property that (Pinv ∧ ¬G) ⇒ Ppost . Step 4: Initialization. For either of the two loop-invariants, the initialization L→ LT L LBL 0 LBR , x→ xT xB , and b → bT bB , where LT L is 0 × 0, and xT , bT have 0 elements, has the property that it sets the variables in a state where the loop-invariant holds. Step 5: Progressing through the operands.