### 5.1.125 qrds

`Syntax:`
  `qrds (` matrix_expression`,` number_expression`,` number_expression`,` number_expression`)`
`Type:`
  list
`Purpose:`
  computes all eigenvalues with multiplicities of the given matrix by performing the numeric QR double shift algorithm involving Hessenberg form and householder transformations.
  This method expects the ground field to be the complex numbers, and all matrix entries to be real numbers, i.e., elements of this ground field with the imaginary part equal to zero.
  If the algorithm works, then it returns a list with two entries which are again lists of the same size:
  _[1][i] is the i-th mutually distinct eigenvalue that was found,
  _[2][i] is the (int) multiplicity of _[1][i].
  If the algorithm does not work (due to an ill-posed matrix), a list with the single entry (int)0 is returned.
  The first number argument is used for detection of deflation in the actual QR double shift algorithm. The second number argument is used for ending Heron's iteration whenever square roots are being computed. And the third number argument is used to distinguish between distinct eigenvalues: When the Euclidean distance between two computed eigenvalues is less then this number, then they will be regarded equal, resulting in a higher multiplicity of the corresponding eigenvalue. (A good choice for all three number arguments is a small value like e.g. 10^(-100).)
`Example:`

        ring r=(complex,50),(dummy),dp;
        matrix A[3][3]=-10,37,-5,-14,51,-10,-29,99,-18;
        bigint b = bigint(10)^100; number t = 1/b;
        list L=qrds(A,t,t,t); L;
      ==> [1]:
      ==>    [1]:
      ==> (3+i*2)
      ==>    [2]:
      ==> (3-i*2)
      ==>    [3]:
      ==> 17
      ==> [2]:
      ==>    [1]:
      ==>       1
      ==>    [2]:
      ==>       1
      ==>    [3]:
      ==>       1

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