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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.

Computing for type matrix<double>

bool  matrix::ReduceSymmetricToTridiagonal(
   matrix&         B,            // tridiagonal matrix
   matrix&         reflect_q,    // q-reflectors
   vector&         tau_q         // scalar factors of the elementary reflectors Q
   );

Computing for type matrix<float>

bool  matrix::ReduceSymmetricToTridiagonal(
   matrixf&        B,            // tridiagonal matrix
   matrixf&        reflect_q,    // q-reflectors
   vectorf&        tau_q         // scalar factors of the elementary reflectors Q
   );

Computing for type matrix<complex>

bool  matrix::ReduceSymmetricToTridiagonal(
   matrixc&        B,            // tridiagonal matrix
   matrixc&        reflect_q,    // q-reflectors
   vectorc&        tau_q         // scalar factors of the elementary reflectors Q
   );

Computing for type matrix<complexf>

bool  matrix::ReduceSymmetricToTridiagonal(
   matrixcf&       B,            // tridiagonal matrix
   matrixcf&       reflect_q,    // q and p-reflectors
   vectorcf&       tau_q         // scalar factors of the elementary reflectors Q
   );

Parameters

B

[out]  Symmetric (or Hermitian) tridiagonal matrix.

reflect_q

[out] Upper or lower triangular matrix, it depends on the input matrix A. In upper case the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix B, and the elements above the first superdiagonal, with the array tau_q, represent the orthogonal matrix Q as a product of elementary reflectors; in lower case the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix B, and the elements below the first subdiagonal, with the array tau_q, represent the orthogonal matrix Q as a product of elementary reflectors.

tau_q

[out] Vector of the scalar factors of the elementary reflectors which represent the orthogonal matrix Q.

Return Value

Return true if successful, otherwise false in case of an error.

Note

In upper case the matrix Q is represented as a product of elementary reflectors

    Q = H(n-1) . . . H(2) H(1)

Each H(i) has the form

    H(i) = I - tau * v * v**T

where tau is a scalar, and v is a vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in tau_q(i).

In lower case the matrix Q is represented as a product of elementary reflectors

    Q = H(1) H(2) . . . H(n-1)

Each H(i) has the form

    H(i) = I - tau * v * v**T

where tau is a scalar, and v is a vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in tau_q(i). */

The contents of A on exit (output matrix reflect_q) are illustrated by the following examples with n = 5:

  upper case:                          lower case:
 
  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of B and vi denotes an element of the vector defining H(i).

Matrix Q can be produced with ReflectTridiagonalToQ method.