复旦大学：《Matlab》（ppt实验英文版）Chapter 7 The QR and Cholesky Factorizations

∑a+)=√]-2a+)==a Minimizer ao 0 aa 3 15 1521尸 31 3264 80 0.37037037 0.651851851 polyfit t If we try to fit data point in the least squares sense with a polynomial of degree d, the m-by-(d+1) least squares problems arises

( )  ( )  ( , ) .75 1 1 .2 5 2         =  + −  + −   x x x x dx m m i i i 0, = 0   =        Minimizer             =                80 31 12 7 64 21 32 15 32 15 4 3 * *          =      0.651851851 0.37037037 * *   polyfit If we try to fit data point in the least squares sense with a polynomial of degree d, the m-by-(d+1) least squares problems arises

Matlabs Least Squares Tools Ax-b x∈Rn is equivalent to a transformed problem min x∈R Q4k-(b)2 2Q=00=l, orthogonal cos(6) sin( 0) 2 sin (o)cos(0) The column of an orthogonal matrix define an orthonormal basis

Matlab’s Least Squares Tools min 2 Ax b n x R −  is equivalent to a transformed problem ( ) ( ) 2 min Q A x Q b T T x R n −  QQ Q Q I orthogonal T T = = , 2 2 2 2 Q r r T =       − = sin( ) cos( ) cos( ) sin( )     Q The column of an orthogonal matrix define an orthonormal basis

min Ax-bll= min oa x∈R2 I2x-(o bl ∈R 22 x mn X∈R2‖00 4 12 22C min Ax-b ls b Ca O x∈R2 2 2

( ) ( ) 2 4 3 2 1 2 2 2 1 1 1 1 2 2 2 0 0 0 0 0 min min min 2 2 2              −                  = − = −    c c c c x r x r r Ax b Q A x Q b x R T T x R x R        =            2 1 2 1 22 11 12 0 c c x x r r r 2 4 2 min 2 Ax b 2 AxLs b 2 c3 c x R − = − = + 

A=OR QR factorization problem To find an orthonormal basis for the subspace defined by the columns of a A(2j)=Q(21)*R(,j+Q(2)*R(2,j)+…+Q(;j)*R(, Any column of a is in the span of (Q( 1).Q( n)) 8 1/3-2/3-2/3‖36 2/3-1/32/3015 214 2/32/3 l/3‖00

A = QR QR factorization problem To find an orthonormal basis for the subspace defined by the columns of A A(:, j) = Q(:,1)*R(1, j) + Q(:,2)*R(2, j) ++ Q(:, j)*R( j, j) Any column of A is in the span of {Q(:,1),…,Q(:,n)}                     − − − − =           − − 0 0 0 15 3 6 2 / 3 2 / 3 1/ 3 2 / 3 1/ 3 2 / 3 1/ 3 2 / 3 2 / 3 2 14 2 1 1 8

Rotations The Q in the qr factorization can be computed using a special family of orthogonal matrices that are called rotations G C=coS(O), S=sin( 0) c=x1/√x2+x2,s=x2/√x2+x2 Gx

Rotations The Q in the QR factorization can be computed using a special family of orthogonal matrices that are called rotations. , = cos( ), = sin( )       − = c s s c c s G 2 2 2 2 1 2 2 2 1 1 c = x / x + x ,s = x / x + x y = Gx, y2 = 0