《结构工程中的数学方法》课程教学课件（讲稿）Cholesky_alternative

2AlgebraicproblemsinmatrixformSystemsofLinearEquationsCholesky decomposition-alternative representationeliminationofsquarerootsA=L*.D.L'Twith:>main diagonal of L* consists of ones only(predefined)D being a diagonal matrix0福0000D0国O00solutionofthesystemA.x=bLDy=b=y=LT.x=y=x61dMichael Beer, Engineering Mathematics

Michael Beer, Engineering Mathematics Systems of Linear Equations Cholesky decomposition − alternative representation 2 Algebraic problems in matrix form 61d ● elimination of square roots * *T A L DL = ⋅⋅ with: » main diagonal of L* consists of ones only (predefined) » D being a diagonal matrix * 21 * * * i1 ij *** n1 n j n n 1 1 0 0 . . 0 l 1 0 . 1 0 l . l 1 0 0 1 0 l . l . l 1 −     =   L       1 i n d 0 . 0 . 0 0 0 0 d 0 . 0 0 . 0 0 0 0 0 . 0 . 0 d     =   D        ● solution of the system Ax b ⋅ = *T Lxy x ⋅= ⇒ * L Dy b y ⋅⋅= ⇒ ⇒

2AlgebraicproblemsinmatrixformSystems of Linear EquationsCholeskydecomposition-introductory example(cont'd)alternative representation,determination of L* and Dd,=1.25-0.502.2.50=az2-l2dd, = 2.50 = al1-1-0)(a32 -512,d,)1.25221[210.625d2.50d.0as1①232.50d.30100100100.50-1.612.50701.250d01.251.25-1.0L.D:[,d,d2=AQ0.6]I5,d,0-1.0[5id,00[2.5000d, = 0.6 - 0 -(-1.6) .0.625 = a33 -I512d, -I52d,D0.625三00-161eMichaelBeer,Engineering Mathematics

alternative representation, determination of L* and D Systems of Linear Equations 2 Algebraic problems in matrix form 61e ● Cholesky decomposition − introductory example (cont'd) 1 * * 21 1 2 * * 31 1 32 2 3 d 00 ld d 0 ld ld d     ⋅ =       L D 2.50 0 0 0 0.625 0 0 01     = −   D * * 21 31 * *T 32 1 l l 0 1 l 0 0 1     =   L 2.5 1.25 0 1.25 1.25 1.0 0 1.0 0.6     − = −   A d 2.50 a 1 = = 11 * 21 21 1 1.25 a l 2.50 d = = * 31 31 1 0 a l 2.50 d = = ( ) ( ) * * * 32 32 31 21 1 2 1 1 l 1 0 a lld 0.625 d = −− = − 2 * 2 d 1.25 0.50 2.50 a l d 2 = − ⋅ =− 22 21 1 ( ) 2 *2 *2 d 0.6 0 1.6 0.625 a l d l d 3 = − −− ⋅ = − − 33 31 1 32 2 * 1 00 0.5 1 0 0 1.6 1     =     −   L Michael Beer, Engineering Mathematics

2AlgebraicproblemsinmatrixformSystemsof LinearEquationsCholesky decomposition-introductory example(cont'd)solution of the system0000d.[2.5000d,1.25L*.D=5,d0.6250-1-1LidI5,d.d》forwardsubstitutionL.D.y=bUY00[2.50y25.0y, =10.001.2525.00.625Y2Yz=20.00-140.0y3 =20.0-1Y3》backward substitutionL*T.x=y=X010.510.0X, =-16.0Xi020.01-1.6X2X, = 52.00020.01X,=20.0X.61fMichaelBeer,EngineeringMathematics

Systems of Linear Equations 2 Algebraic problems in matrix form 61f Cholesky decomposition − introductory example (cont'd) ● solution of the system » forward substitution » backward substitution * L Dy b y ⋅⋅= ⇒ 1 2 3 2.50 0 0 y 25.0 1.25 0.625 0 y 25.0 0 1 1 y 40.0           ⋅ =       −− −          y 10.0 1 = y 20.0 2 = y 20.0 3 = 1 2 3 1 0.5 0 x 10.0 0 1 1.6 x 20.0 0 0 1 x 20.0           −⋅ =                *T Lxy x ⋅= ⇒ x 16.0 1 = − x 52.0 2 = x 20.0 3 = 1 * * 21 1 2 * * 31 1 32 2 3 d 0 0 2.50 0 0 l d d 0 1.25 0.625 0 ld ld d 0 1 1         ⋅ = =        − −      L D Michael Beer, Engineering Mathematics