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# CNC 183 F .ppt

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## CNC 183 F .pptPresentation Transcript

• A Computationally Simplified Numerical Algorithm for Evaluating a Determinant Authors : Dr. S.N. Sivanandam* and Ramani Bai. V** *Professor, PSG Institute of Technology, Coimbatore **Associate Professor, Saintgits College of Engineering, Kottayam
• Determinants
• A Determinant is a uniquely defined intrinsic number associated with a square matrix.
• Determinants are useful in eigenvalue problems and differential equations.
• Determinant shows the consistency of a system of linear equations.
Ramani Bai.V, CNC 183 F 10 March 2011
• The New Proposed Numerical Method
• Utilizes the basic properties of Determinants
• Extracts column vectors using simple row reduction techniques
• Converts an Ill-conditioned Matrix into a well-conditioned matrix
• The efficiency of New Algorithm
10 March 2011 Ramani Bai.V, CNC 183 F
• Extracting Column Vectors
• The proposed method reduces the determinant A into a sequence of n column vectors C 0 , C 1 , C 2 … C n-1.
• The value of determinant A is obtained by multiplying all the column vectors as ΔA = C 0 x C 1 x C 2 x …x C n-1 i.e,
• ΔA = A 0 [1:n,1:1] x A 1 [1:(n-1),2:2] x A 2 [1:(n-2),3:3] x … x A n-1 [1:1,n:n]
10 March 2011 Ramani Bai.V, CNC 183 F
• Extracting Column Vectors 10 March 2011 Ramani Bai.V, CNC 183 F
• Extracting Column Vectors 10 March 2011 Ramani Bai.V, CNC 183 F
• Extracting Column Vectors 10 March 2011 Ramani Bai.V, CNC 183 F
• Extracting Column Vectors 10 March 2011 Ramani Bai.V, CNC 183 F
• Extracting Column Vectors 10 March 2011 Ramani Bai.V, CNC 183 F
• Evaluating the Determinant 10 March 2011 Ramani Bai.V, CNC 183 F
• Ill-Conditioned into Well-Conditioned 10 March 2011 Ramani Bai.V, CNC 183 F
• Numerical Algorithm for Proposed Method 10 March 2011 Ramani Bai.V, CNC 183 F
• Numerical Algorithm for Proposed Method 10 March 2011 Ramani Bai.V, CNC 183 F
• Analysis of Algorithm
• Algorithm ProposedMethod_EvaluateDeterminant requires approximately (n-1)+(n-1) 2 divisions and 2n+(n-1) subtractions only.
• Algorithm IllConditionedDeterminant requires only n+n 2 additions.
• For evaluating the determinant, the step 4 in algorithm ProposedMethod_EvaluateDeterminant requires approximately 2n+(n-1) multiplications only.
10 March 2011 Ramani Bai.V, CNC 183 F
• Numerical Illustrations 10 March 2011 Ramani Bai.V, CNC 183 F
• Numerical Illustrations 10 March 2011 Ramani Bai.V, CNC 183 F
• Numerical Illustrations 10 March 2011 Ramani Bai.V, CNC 183 F
• Conclusion
• Reliable in evaluating any determinant with a chosen accuracy of 10 -12 .
• Does not use partial pivoting like Gauss reduction method, the complexity involved in this algorithm is reduced compared to other methods.
• Most suitable for ill-conditioned determinants.
• The procedure is suitably modified to evaluate the determinant having complex values.
10 March 2011 Ramani Bai.V, CNC 183 F
• 10 March 2011 Ramani Bai.V, CNC 183 F