(i)  Matrix Inversion 
(ii)  Matrix Factorizations 
(iii)  Matrix Arithmetic and Manipulation 
(iv)  Matrix Functions 
(i)  Nonsingular square matrices of order If
It is worth noting that if 

(ii)  General real rectangular matrices.
A real matrix

(a)  Nonsingular square matrices of order This chapter describes techniques for inverting a general real matrix
Routines are given for calculating the approximate inverse, that is solving the linear equations just once, and also for obtaining the accurate inverse by successive iterative corrections of this first approximation. The latter, of course, are more costly in terms of time and storage, since each correction involves the solution of
Despite the extra work of the ‘corrected’ inverse routines they are superior to the ‘approximate’ inverse routines. A correction provides a means of estimating the number of accurate figures in the inverse or the number of ‘meaningful’ figures relating to the degree of uncertainty in the coefficients of the matrix.
The residual matrix
The decision trees for inversion show which routines in
Chapter F04 and
Chapter F07 should be used for the inversion of other special types of matrices not treated in the chapter. 
(b)  General real rectangular matrices
For real matrices F08AEF (DGEQRF)
and F01QJF return
F01BLF uses the
F08KBF (DGESVD) and F08KPF (ZGESVD)
compute the singular value decomposition as described in Section 2 for real and complex matrices respectively. If 
Is 
_ yes 
Is 
_ yes 
see Tree 2 
  no  

  see Tree 3  
no  

see Tree 4 
Is 
_ yes 
See Note 1.  
no  

Is 
_ yes 
Is 
_ yes 
Do you want guaranteed accuracy? (See Note 2)  _ yes 
F01ABF 
    no  

    Is one triangle of 
_ yes 
F07GDF and F07GJF  
    no  

    F01ADF or F07FDF and F07FJF  
  no  

  Is one triangle of 
_ yes 
F07PDF and F07PJF  
  no  

  F07MDF and F07MJF  
no  

Is 
_ yes 
Is 
_ yes 
F07UJF  
  no  

  F07TJF  
no  

Do you want guaranteed accuracy? (See Note 2)  _ yes 
F04AEF  
no  

F07ADF and F07AJF 
Is 
_ yes 
See Note 1.  
no  

Is 
_ yes 
Is 
_ yes 
Is one triangle of 
_ yes 
F07GRF and F07GWF 
    no  

    F07FRF and F07FWF  
  no  

  Is one triangle 
_ yes 
F07PRF and F07PWF  
  no  

  F07MRF and F07MWF  
no  

Is 
_ yes 
Is one triangle of 
_ yes 
F07QRF and F07QWF  
  no  

  F07NRF and F07NWF  
no  

Is 
_ yes 
Is 
_ yes 
F07UWF  
  no  

  F07TWF  
no  

F07ANF or F07ARF and F07AWF 
Is 
_ yes 
Is 
_ yes 
Is 
_ yes 
F01RJF and F01RKF 
    no  

    F08ASF and F08AUF or F08ATF  
  no  

  F08KPF  
no  

Is 
_ yes 
Is 
_ yes 
F01QJF and F01QKF  
  no  

  F08AEF and F08AGF or F08AFF  
no  

Is 
_ yes 
F08KBF  
no  

Is reliability more important than efficiency?  _ yes 
F08KBF  
no  

F01BLF 