1.9 KiB
1.9 KiB
Reductions
Functions designed to transform matrices into special forms using orthogonal transformations. These methods are widely used in numerical computations, such as spectral analysis, eigenvalue problems, and singular value decomposition (SVD) of matrices.
These functions enable efficient matrix transformations essential for solving linear algebra problems, including computing Singular Value Decomposition (SVD) and finding eigenvalues of symmetric matrices. The use of LAPACK functions ensures high performance and computational accuracy.
| Function | Action |
|---|---|
| ReduceToBidiagonal | Reduces a general real or complex m-by-n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. LAPACK function GEBRD . |
| ReflectBidiagonalToQP | Generates orthogonal matrices Q and PT (or PH for complex types) determined by ReduceToBidiagonal method when reducing a real or complex matrix A to bidiagonal form: A = Q * B * PT. Q and PT are defined as products of elementary reflectors H(i) or G(i) respectively. LAPACK functions ORGBR , UNGBR . |
| ReduceSymmetricToTridiagonal | Reduces a real symmetric or complex Hermitian matrix A to trdiagonal form B by an orthogonal similarity transformation: Q**T * A * Q = B. LAPACK functions SYTRD , HETRD . |
| ReflectTridiagonalToQ | Generates orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by ReduceSymmetricToTridiagonal |
| ReduceToHessenberg | Reduces a real or complex general n-by-n matrix A to upper Hessenberg form B by an orthogonal similarity transformation: Q**T * A * Q = H. LAPACK function GEHRD . |
| ReflectHessenbergToQ | Generates orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by ReduceToHessenberg : Q = H(1) H(2) . . . H(n-1). LAPACK function ORGHR . |