Matrix Chain Multiplication Example Step By Step Ppt

10 Sep, 2021

We can multiply the chain of matrices by following those conditions in these ways. Problem Statement What is the number of possible parenthesizations.

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Example of Finding the Multiplication Sequence.

Matrix chain multiplication example step by step ppt. Int matOrderint array int n int minMulnn. The matrices have size 4 x 10 10 x 3 3 x 12 12 x 20 20 x 7. For k 1 to q 5.

Example of Matrix Chain Multiplication. 211 -4-2 -16 18 32. I minMulii 0.

ABCD - This is a 2x2 multiplied by a 2x1. Multiplications for BC 10x50x20 10000 creating a 10x20 answer matrix multiplications for ABC 2x10x20 400 Total multiplications 10000. N length p-1 Where n is the total number of elements And length p 5 n 5 - 1 4 n 4 Now we construct two tables m and s.

Summation notation for a matrix squared Let A be an N N matrix. For any optimal multiplication sequence at the last step youare multiplyingtwo matrices and for some. Consider three matrices A10 100 B100 5 and C5 50 There are 2 ways to parenthesize ABC D10 5 C5 50 AB 1010055000 scalar multiplications DC 10550 2500 scalar multiplications ABC.

Matrix multiplication Let A a ij and B b ij be N N matrices. Then A2 ij XN k1 A ikA kj XN k1 a ika kj. The minimum number of multiplications are obtained by putting parenthesis in following way A BCD -- 203010 402010 401030 Input.

We will show that the way we group matrices when multiplying A B C matters. In the example of Figure 161 the call MATRIX-CHAIN-MULTIPLYA s 1 6 computes the matrix-chain product according to the parenthesization A 1 A 2 A 3A 4 A 5A 6. ABCD - This is a 2x4 multiplied by a 4x1 so 2x4x1 8 multiplications plus whatever work it will take to multiply BCD.

Let us proceed with working away from the diagonal. Ci j Ci j Ai k Bk j 6return C Scalar multiplication in line 5 dominates time to compute CNumber of scalar multiplications pqr 11- Matrix-chain Multiplicationcontd Example. A 1A 2A 3A 4 A 23 A 2A 3 25000 mults result is 100 by 50 A 13 A 1A 23 50000 mults result is 10 by 50 A 14 A 13A 4 500 mults results is 10 by Total is 75500 A 1A 2A 3A 4 A 23 A 2A.

I int j ilength-1. Let the input 4 matrices. Matrix Chain Multiplication Lets get back to our example.

We need to compute M ij 0 i j 5. 75000 AE 1010050 50000 scalar multiplications Total. We compute the optimal solution for the product of 2 matrices.

The matrices have size 4 x 10 10 x 3 3 x. Matrix-Matrix Multiplication An Example. Computing the Optimal Solution Example FASTEST-WAYa t e x n FASTEST-WAYa t e x n cont 4.

30000 There are 4 matrices of dimensions 10x20 20x30 30x40 and 40x30. Let A be a 2x10 matrix Let B be a 10x50 matrix Let C be a 50x20 matrix Consider computing ABC. Let the input 4 matrices be A B C and D.

We are given the sequence 4 10 3 12 20 and 7. Holds the number of scalar multiplication needed for int i1. The Chain Matrix Multiplication Problem Given dimensions corresponding to matr 5 5 5 ix sequence.

Construct an Optimal Solution Matrix-Chain Multiplication MATRIX-MULTIPLYA B Matrix-Chain Multiplication Example Matrix-Chain Multiplication. Example A 1 is 10 by 100 matrix A 2 is 100 by 5 matrix A 3 is 5 by 50 matrix A 4 is 50 by 1 matrix A 1A 2A 3A 4 is a 10 by 1 matrix. That is 5 Example 5.

We are given the sequence 4 10 3 12 20 and 7. Include using namespace std. Consider three matrices A10 100 B100 5 and C5 50 There are 2 ways to parenthesize ABC D10 5 C5 50 AB 1010055000 scalar multiplications DC 10550 2500 scalar multiplications A BC A10 100 E100 50 BC 10055025000 scalar multiplications Total.

Length find the chain length starting from 2 for int i1. The product matrix is A B AB with elements AB ij XN k1 a ikb kj. P 10 20 30 40 30 Output.

2 then X MATRIX-CHAIN-MULTIPLYA s i si j 3 Y MATRIX-CHAIN-MULTIPLYA s si j 1 j 4 return MATRIX-MULTIPLYX Y 5 else return A i. For multiplication with 1 matrix cost is 0 for int length2. Following that we multiply the elements along the first row of matrix A with the corresponding elements down the second column of matrix B then add the results.

We know M i i 0 for all i. Example of Matrix Chain Multiplication Example. Length of array P number of elements in P length p 5 From step 3 Follow the steps in Algorithm in Sequence According to Step 1 of Algorithm Matrix-Chain-Order Step 1.

P 10 20 30 40 30 Output. The minimum number of multiplications are obtained by putting parenthesis in following way A BCD -- 203010 402010 401030 Input. This gives us the answer well need to put in the first row second column of the answer matrix.

The Chain Matrix Multiplication Problem De nition Chain matrix multiplication problem Given dimensions p 0p 1p n corresponding to matrix sequence A 1 A 2 A n in which A i has dimension p i 1 p i determine the multiplication sequencethatminimizesthe number ofscalar multiplicationsin computing A 1A 2 A n. 30000 There are 4 matrices of dimensions 10x20 20x30 30x40 and 40x30. Matrix Chain Multiplication Consider the case multiplying these 4 matrices.

M1 M2 M3 M4 2x3 3x4 4x5 5x7 M1 M2 M3 M4 M1 M2 M3 M4 M1 M 2 M 3 M4 M1 M 2 M 3 M 4 M1 M2 M3 M4 220 134 266 160 207 Numbers of the rightmost side is number of total scalar multiplication. 4 0 2 1 3 1 1 0 12 0 2 0 10 Matrix-Matrix Multiplication Matrix-Matrix Multiplication Matrix-Matrix. Pre-multiplication of a matrix by a vector Let A be an NN matrix and let π be an N1 column vector.

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