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AlmostQR    --    QR decomposition of a matrix


Syntax 
AlmostQR(M: MAT): RECORD

Description
This function computes the decomposition of the matrix into an orthogonal and an upper triangular matrix with 1 on the diagonal. [orthogonal meaning that QT * Q is a diagonal matrix]

The auxiliary (possibly slow!) function Mat.SimplifySquareFactorsInAQR modifies Q and R in the decomposition so that the entries of the diagonal matrix QT * Q are squarefree rationals.

Example 
/**/ M := matrix([ [4, -2, 3],  [3, 2, -2],  [0, 0, 3] ]);
/**/ Dec := AlmostQR(M);
/**/ indent(Dec);
record[
  Q := matrix(QQ,
 [[4,  -42/25,  0],
  [3,  56/25,  0],
  [0,  0,  3]]),
  R := matrix(QQ,
 [[1,  -2/25,  6/25],
  [0,  1,  -17/14],
  [0,  0,  1]])
]

/**/  $mat.SimplifySquareFactorsInAQR(ref Dec);
/**/  indent(Dec);
record[
  Q := matrix(QQ,
 [[4/5,  -3/5,  0],
  [3/5,  4/5,  0],
  [0,  0,  1]]),
  R := matrix(QQ,
 [[5,  -2/5,  6/5],
  [0,  14/5,  -17/5],
  [0,  0,  3]]),
  SqDiag := [1,  1,  1]
]

See Also