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A(1、m)とB(1、m)の演算A \ Bの結果はどうなりますか?

マニュアルには次のように書かれています。

A\B returns a least-squares solution to the system of equations A*x= B.

つまり、x = inv(A'* A)* A'* B?ただし、行列A'*Aは特異です...

仮定しましょう:

A=[1 2 3]
B=[6 7 6]
A\B

0         0         0
0         0         0
2.0000    2.3333    2.0000

MLSを使用した場合:

C = inv (A'*A)   singular matrix
C = pinv(A'*A)

0.0051    0.0102    0.0153
0.0102    0.0204    0.0306
0.0153    0.0306    0.0459

D= C*A'*B

0.4286    0.5000    0.4286
0.8571    1.0000    0.8571
1.2857    1.5000    1.2857

したがって、結果A \ Bとinv(A'* A)*A'*Bは異なります。

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2 に答える 2

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My MATLAB (R2010b) says quite a lot about what A\B does:

mldivide(A,B) and the equivalent A\B perform matrix left division (back slash). A and B must be matrices that have the same number of rows, unless A is a scalar, in which case A\B performs element-wise division — that is, A\B = A.\B.

If A is a square matrix, A\B is roughly the same as inv(A)*B, except it is computed in a different way. If A is an n-by-n matrix and B is a column vector with n elements, or a matrix with several such columns, then X = A\B is the solution to the equation AX = B. A warning message is displayed if A is badly scaled or nearly singular.

If A is an m-by-n matrix with m ~= n and B is a column vector with m components, or a matrix with several such columns, then X = A\B is the solution in the least squares sense to the under- or overdetermined system of equations AX = B. In other words, X minimizes norm(A*X - B), the length of the vector AX - B. The rank k of A is determined from the QR decomposition with column pivoting. The computed solution X has at most k nonzero elements per column. If k < n, this is usually not the same solution as x = pinv(A)*B, which returns a least squares solution.

mrdivide(B,A) and the equivalent B/A perform matrix right division (forward slash). B and A must have the same number of columns.

If A is a square matrix, B/A is roughly the same as B*inv(A). If A is an n-by-n matrix and B is a row vector with n elements, or a matrix with several such rows, then X = B/A is the solution to the equation XA = B computed by Gaussian elimination with partial pivoting. A warning message is displayed if A is badly scaled or nearly singular.

If B is an m-by-n matrix with m ~= n and A is a column vector with m components, or a matrix with several such columns, then X = B/A is the solution in the least squares sense to the under- or overdetermined system of equations XA = B.

于 2012-11-14T15:22:12.133 に答える
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x = inv (A'*A)*A'*B過剰に決定されたシステム (つまり、これらの状況で ; を含むマトリックスAとしての機能は可逆)に適しています。n x mn>mA'A

あなたの場合、決定されていないシステムがあります。


では、何が起こるでしょうか?

私の意見では、少なくともあなたの場合は確認できますが

あなたがそうするとき、A\Bmatlabは通常の最小二乗法とは逆の意味で最適化問題を解きます。

 X = argmin_{X \in S} ||X||,

Sソリューションのセットはどこにありますか。つまり、最小の L^2 ノルムを持つシステムの解が得られます。(少なくともあなたの場合は、手で問題を処理できると考えてください)。

于 2012-11-14T15:12:54.187 に答える