Tests for the Least Squares Driver Routines

In the SLS path, driver routines are tested for computing solutions to over- and underdetermined, possibly rank-deficient systems of linear equations 93#93 (16#16 is 3#3-by-4#4). For each test matrix type, we generate three matrices: One which is scaled near underflow, a matrix with moderate norm, and one which is scaled near overflow. The SGELS driver computes the least-squares solutions (when 94#94) and the minimum-norm solution (when 62#62) for an 3#3-by-4#4 matrix 16#16 of full rank. To test SGELS, we generate a diagonally dominant matrix 16#16, and for 95#95 and 96#96, we

• generate a consistent right-hand side 97#97 such that 98#98 is in the range space of 67#67, compute a matrix 98#98 using SGELS, and compute the ratio
  
  99#99
  
  

• If 67#67 has more rows than columns (i.e. we are solving a least-squares problem), form 100#100, and check whether 74#74 is orthogonal to the column space of 16#16 by computing
  
  101#101
  
  

• If 67#67 has more columns than rows (i.e. we are solving an overdetermined system), check whether the solution 98#98 is in the row space of 67#67 by scaling both 98#98 and 67#67 to have norm one, and forming the QR factorization of 102#102 if 96#96, and the LQ factorization of 103#103 if 95#95. Letting 104#104 in the first case, and 105#105 in the latter, we compute
  
  106#106
  
  

The SGELSX, SGELSY, SGELSS and SGELSD drivers solve a possibly rank-deficient system 93#93 using a complete orthogonal factorization (SGELSX or SGELSY) or singular value decomposition (SGELSS or SGELSD), respectively. We generate matrices 16#16 that have rank 107#107 or rank 108#108 and are scaled to be near underflow, of moderate norm, or near overflow. We also generate the null matrix (which has rank 109#109). Given such a matrix, we then generate a right-hand side 97#97 which is in the range space of 16#16.

In the process of determining 98#98, SGELSX (or SGELSY) computes a complete orthogonal factorization 110#110, whereas SGELSS (or SGELSD) computes the singular value decomposition 111#111.

• If 52#52 are the true singular values of 16#16, and 78#78 are the singular values of 85#85, we compute
  
  112#112
  
  
  for SGELSX (or SGELSY), and
  
  113#113
  
  
  for SGELSS (or SGELSD).

• Compute the ratio
  
  99#99
  
  

• If 114#114, form 100#100, and check whether 74#74 is orthogonal to the column space of 16#16 by computing
  
  115#115
  
  

• If 116#116, check if 98#98 is in the row space of 16#16 by forming the LQ factorization of 103#103. Letting 105#105, we return
  
  106#106