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Chapter 9.2: Matrix-chain Multiplication A5 A6 10 × 20 20 × 25 matrix imension 1 2 0 A j 3 A₁ A2 30 × 35 35 × 15 4 0 6 15,750 2,625 A₂ 7,875 4,375 m 5 2 11,875 10,500 0 15,125 A₂ 1 9,375 7,125 5,375 4 2,500 3,500 750 A3 15 × 5 А₁ 3 m[2,2]+m[3,5],+p, P₂P = 0 + 2,500 + 35·15-20 = 13,00C m[2,5] = min m[2,3]+m[4,5],+p₁ p²p² = 2,625 + 1,000 + 35.5.20 = 7,1 m[2,4]+m[5,5],+p₁P₁P² = 4,375 + 0 + 35∙10-20 = 11,375 i 0 A4 5 × 10 1,000 5,000 As 5 0 A 6 Definitions n Want: A, A₂...A (costs m[1, Dimensions: {P. P} A₁ =A₁A₁A₁+2A, "i+1 m[i, j] = min cost for A i:j =7,125 ..... This approach gives us the minimum cost (minimum number of pairwise matrix element multiplications), but not the parenthecization itself. As with CUT-ROD, we can track this separately in a matrix we'll call S.

Chapter 9.2: Matrix-chain Multiplication A5 A6 10 × 20 20 × 25 matrix imension 1 2 0 A j 3 A₁ A2 30 × 35 35 × 15 4 0 6 15,750 2,625 A₂ 7,875 4,375 m 5 2 11,875 10,500 0 15,125 A₂ 1 9,375 7,125 5,375 4 2,500 3,500 750 A3 15 × 5 А₁ 3 m[2,2]+m[3,5],+p, P₂P = 0 + 2,500 + 35·15-20 = 13,00C m[2,5] = min m[2,3]+m[4,5],+p₁ p²p² = 2,625 + 1,000 + 35.5.20 = 7,1 m[2,4]+m[5,5],+p₁P₁P² = 4,375 + 0 + 35∙10-20 = 11,375 i 0 A4 5 × 10 1,000 5,000 As 5 0 A 6 Definitions n Want: A, A₂...A (costs m[1, Dimensions: {P. P} A₁ =A₁A₁A₁+2A, "i+1 m[i, j] = min cost for A i:j =7,125 ..... This approach gives us the minimum cost (minimum number of pairwise matrix element multiplications), but not the parenthecization itself. As with CUT-ROD, we can track this separately in a matrix we'll call S.

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**Chapter 9.2: Matrix-chain Multiplication** --- **Objective:** To find the optimal way to multiply a chain of matrices, minimizing the computation cost (i.e., number of scalar multiplications). **Matrix Dimensions:** - \( A_1 \): \( 30 \times 35 \) - \( A_2 \): \( 35 \times 15 \) - \( A_3 \): \( 15 \times 5 \) - \( A_4 \): \( 5 \times 10 \) - \( A_5 \): \( 10 \times 20 \) - \( A_6 \): \( 20 \times 25 \) **Definitions:** - **Goal**: Compute \( A_1 A_2 \cdots A_n \) with the minimum cost, denoted as \( m[1,n] \). - **Dimensions**: Defined as \( \{ p_0, \ldots, p_n \} \). - **Matrix Product**: \( A_{i,j} = A_i A_{i+1} \cdots A_j \). - The **minimum cost** for multiplying matrices from \( A_i \) to \( A_j \) is \( m[i,j] \). **Explanation of Table and Diagram:** The triangular matrix shows the intermediate calculations needed to derive the minimum multiplication cost: - **Indices** \( i \) and \( j \): Index matrices in the order they are multiplied. - **Colors**: Indicate various calculations at different matrix boundaries and configurations. For example, \( m[2,5] \) is derived from considering different ways of parenthesizing the multiplication of matrices \( A_2 \) to \( A_5 \): - **Calculation Breakdown**: - \( m[2,2] + m[3,5] + p_1p_2p_5 = 0 + 2500 + 35 \times 15 \times 20 = 13,000 \) (shaded in red) - \( m[2,3] + m[4,5] + p_1p_3p_5 = 2,625 + 1,000 + 35 \times 5 \times 20 = 7,125 \) (shaded in blue) - \( m[2,4] + m[
Transcribed Image Text:**Chapter 9.2: Matrix-chain Multiplication** --- **Objective:** To find the optimal way to multiply a chain of matrices, minimizing the computation cost (i.e., number of scalar multiplications). **Matrix Dimensions:** - \( A_1 \): \( 30 \times 35 \) - \( A_2 \): \( 35 \times 15 \) - \( A_3 \): \( 15 \times 5 \) - \( A_4 \): \( 5 \times 10 \) - \( A_5 \): \( 10 \times 20 \) - \( A_6 \): \( 20 \times 25 \) **Definitions:** - **Goal**: Compute \( A_1 A_2 \cdots A_n \) with the minimum cost, denoted as \( m[1,n] \). - **Dimensions**: Defined as \( \{ p_0, \ldots, p_n \} \). - **Matrix Product**: \( A_{i,j} = A_i A_{i+1} \cdots A_j \). - The **minimum cost** for multiplying matrices from \( A_i \) to \( A_j \) is \( m[i,j] \). **Explanation of Table and Diagram:** The triangular matrix shows the intermediate calculations needed to derive the minimum multiplication cost: - **Indices** \( i \) and \( j \): Index matrices in the order they are multiplied. - **Colors**: Indicate various calculations at different matrix boundaries and configurations. For example, \( m[2,5] \) is derived from considering different ways of parenthesizing the multiplication of matrices \( A_2 \) to \( A_5 \): - **Calculation Breakdown**: - \( m[2,2] + m[3,5] + p_1p_2p_5 = 0 + 2500 + 35 \times 15 \times 20 = 13,000 \) (shaded in red) - \( m[2,3] + m[4,5] + p_1p_3p_5 = 2,625 + 1,000 + 35 \times 5 \times 20 = 7,125 \) (shaded in blue) - \( m[2,4] + m[
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