《结构工程中的数学方法》课程教学课件（讲稿）intro_Cholesky

2AlgebraicproblemsinmatrixformSystems of Linear EquationsCholeskydecomposition-introductoryexample(fromslide33a)matrixofcoefficientswith:a=2.50m,b=1.25m,g=32.0kN/m02.51.250V2.50家A=1.251.25-1.0NO.62500.60-1.00·Cholesky decomposition:A=L.LT13111212.50== LT132a-C22001330(2.5)1.25O01.251.25-1.0=A70-1.00.61.25221V2.50I1003y131I11V2.5061aMichaelBeer,EngineeringMathematics

Michael Beer, Engineering Mathematics matrix of coefficients with: Systems of Linear Equations Cholesky decomposition − introductory example (from slide 33a) 2 Algebraic problems in matrix form 61a ● a 2.50 m, b 1.25 m, q 32.0 kN/m = = = 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     = − −   A ● Cholesky decomposition: 11 21 31 T 22 32 33 l l l 0 l l 0 0 l     =   L 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     − = −   A T A LL = ⋅ 11 21 22 31 32 33 l 00 ll0 lll     =       L 11 11 l 2.50 a = = 21 21 11 1.25 a l 2.50 l = = 31 31 11 0 a l 2.50 l = = 2.50 0 0 0.625 0 0     =   L   

2AlgebraicproblemsinmatrixformSystemsof LinearEquationsCholeskydecomposition-introductoryexample(cont'd)matrixofcoefficientswith:a=2.50m,b=1.25m,g=32.0kN/m02.5 1.250V2.50CA=1.251.25-1.0JO.6250L=0 -1.00.60·Choleskydecomposition:A=L.LT3:=LT0=V1.25-0.62522320033001.25001.251.25-1.0=A00.6-1.031353300V2.501-1-0.V0.625)=a -,1V0.625037V0.625V0.6250V1.661bMichael Beer,EngineeringMathematics

matrix of coefficients with: Systems of Linear Equations Cholesky decomposition − introductory example (cont'd) 2 Algebraic problems in matrix form 61b ● a 2.50 m, b 1.25 m, q 32.0 kN/m = = = 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     = − −   A ● Cholesky decomposition: 11 21 31 T 22 32 33 l l l 0 l l 0 0 l     =   L 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     − = −   A T A LL = ⋅ 11 21 22 31 32 33 l 00 ll0 lll     =       L 2.50 0 0 0.625 0 0     =   L    32 ( ) ( 32 31 21 ) 22 1 1 l 1 0 0.625 a l l 0.625 l = −− ⋅ = − 2 22 22 21 l 1.25 0.625 a l = − =− 2.50 0 0 0.625 0.625 0 0 1.6     = −   L  Michael Beer, Engineering Mathematics

2AlgebraicproblemsinmatrixformSystemsofLinearEquationsCholeskydecomposition-introductoryexample(cont'd)matrixofcoefficientswith:a=2.50m,b=1.25m,g=32.0kN/m02.51.25V2.5000A=1.251.25-1.0V0.625V0.6250L=0-1.00.6/1.60.Choleskydecomposition:A=L.LT332=0.6-0-1.6=a33-132-1=LT0S13200'3301.2502.501.25-1.0=A1.2500.6-1.00V2.5000V0.625V0.625L=1.60reason:det(A)<0V-161cMichael Beer,Engineering Mathematics

matrix of coefficients with: Systems of Linear Equations Cholesky decomposition − introductory example (cont'd) 2 Algebraic problems in matrix form 61c ● a 2.50 m, b 1.25 m, q 32.0 kN/m = = = 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     = − −   A ● Cholesky decomposition: 11 21 31 T 22 32 33 l l l 0 l l 0 0 l     =   L 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     − = −   A T A LL = ⋅ 11 21 22 31 32 33 l 00 ll0 lll     =       L 2.50 0 0 0.625 0.625 0 0 1.6 1     = − −   L 2.50 0 0 0.625 0.625 0 0 1.6     = −   L  2 2 2 33 33 31 32 l 0.6 0 1.6 a l l = −− = − − reason: det(A) < 0 Michael Beer, Engineering Mathematics