How can I find out what algorithm mldivide is using?

As I understand it, MATLAB's mldivide function uses a variety of algorithms to solve linear systems, performing various checks on the matrix involved to see what algorithm is applicable. These details are hidden from the user.

I recall that I have seen that there is some way to make MATLAB output the details of this procedure at runtime, where it prints out the result of the checks and the algorithm it decided on (LU, Cholesky, something from LAPACK even for sparse systems?). I can't find this information again despite my best attempts at searching these forums, the documentation, and the web. I know I have seen it somewhere. I was hoping someone here knows how to do this.

Failing this, can anyone tell me what algorithm mldivide would choose for the following? I have a pentadiagonal system in a sparse (2L x 2L) matrix for some whole number L

S = spdiags(myDiagonals,-2:2,2*L,2*L);

And a sparse right hand side with four nonzero entries.

b = spalloc(2*L,1,4);

Both contain complex entries in general and S it isn't symmetric or Hermitian or Toeplitz, but I'm not sure what other qualifiers might cause MATLAB to choose a specific algorithm. I calculate the inverse by

S\b

I appreciate any insights into what mldivide is doing in this case or how to get MATLAB to output the details of it's choice of algorithm. Thanks.

[Jess]

You can tell MATLAB to output some of this information by running the following command before calling mldivide():

spparms('spumoni',2)

Answer 1

[Grzegorz Knor]

See spparms function. For example:

A = sparse(diag(rand(5,1)))
spparms('spumoni',2)
A\rand(5,1)

produces:

sp\: bandwidth = 0+1+0.
sp\: is A diagonal? yes.
sp\: do a diagonal solve.

Or:

A = sparse(rand(10).*round(rand(10)-0.2));
spparms('spumoni',2)
A\rand(10,1)

porduces very long and detailed output:

sp\: bandwidth = 7+1+6.
sp\: is A diagonal? no.
sp\: is band density (0.35) > bandden (0.50) to try banded solver? no.
sp\: is A triangular? no.
sp\: is A morally triangular? no.
sp\: is A a candidate for Cholesky (symmetric, real positive diagonal)? no.
sp\: use Unsymmetric MultiFrontal PACKage with automatic reordering.
UMFPACK V5.4.0 (May 20, 2009), Control:
    Matrix entry defined as: double
    Int (generic integer) defined as: UF_long

      0: print level: 2
      1: dense row parameter:    0.2
          "dense" rows have    > max (16, (0.2)*16*sqrt(n_col) entries)
      2: dense column parameter: 0.2
          "dense" columns have > max (16, (0.2)*16*sqrt(n_row) entries)
      3: pivot tolerance: 0.1
      4: block size for dense matrix kernels: 32
      5: strategy: 0 (auto)
      6: initial allocation ratio: 0.7
      7: max iterative refinement steps: 2
      12: 2-by-2 pivot tolerance: 0.01
      13: Q fixed during numerical factorization: 0 (auto)
      14: AMD dense row/col parameter:    10
         "dense" rows/columns have > max (16, (10)*sqrt(n)) entries
          Only used if the AMD ordering is used.
      15: diagonal pivot tolerance: 0.001
          Only used if diagonal pivoting is attempted.
      16: scaling: 1 (divide each row by sum of abs. values in each row)
      17: frontal matrix allocation ratio: 0.5
      18: drop tolerance: 0
      19: AMD and COLAMD aggressive absorption: 1 (yes)

      The following options can only be changed at compile-time:
      8: BLAS library used:  Fortran BLAS.  size of BLAS integer: 8
      9: compiled for MATLAB
      10: CPU timer is ANSI C clock (may wrap around).
      11: compiled for normal operation (debugging disabled)
      computer/operating system: Microsoft Windows
      size of int: 4 UF_long: 8 Int: 8 pointer: 8 double: 8 Entry: 8 (in bytes)

sp\: UMFPACK's factorization was successful.
sp\: However, UMFPACK may have failed to produce a correct factorization
     possibly due to aggressive pivot tolerances or because the problem
     is ill-conditioned.
sp\: These tolerances were:
       Pivot Tolerance:           1.00e-01
       Diagonal Pivot Tolerance:  1.00e-03
sp\: We will factor the matrix again but use standard partial pivoting.
sp\: UMFPACK's solve was successful.
UMFPACK V5.4.0 (May 20, 2009), Info:
    matrix entry defined as:          double
    Int (generic integer) defined as: UF_long
    BLAS library used: Fortran BLAS.  size of BLAS integer: 8
    MATLAB:                           yes.
    CPU timer:                        ANSI clock ( ) routine.
    number of rows in matrix A:       10
    number of columns in matrix A:    10
    entries in matrix A:              32
    memory usage reported in:         16-byte Units
    size of int:                      4 bytes
    size of UF_long:                  8 bytes
    size of pointer:                  8 bytes
    size of numerical entry:          8 bytes

      strategy used:                    unsymmetric
      ordering used:                    colamd on A
      modify Q during factorization:    yes
      prefer diagonal pivoting:         no
      pivots with zero Markowitz cost:               1
      submatrix S after removing zero-cost pivots:
          number of "dense" rows:                    0
          number of "dense" columns:                 0
          number of empty rows:                      1
          number of empty columns                    0
          submatrix S not square or diagonal not preserved
      symbolic factorization defragmentations:       0
      symbolic memory usage (Units):                 247
      symbolic memory usage (MBytes):                0.0
      Symbolic size (Units):                         57
      Symbolic size (MBytes):                        0
      symbolic factorization CPU time (sec):         0.00
      symbolic factorization wallclock time(sec):    0.00

      matrix scaled: yes (divided each row by sum of abs values in each row)
      minimum sum (abs (rows of A)):              8.98358e-01
      maximum sum (abs (rows of A)):              3.05977e+00

      symbolic/numeric factorization:      upper bound               actual      %
      variable-sized part of Numeric object:
          initial size (Units)                     199                  189    95%
          peak size (Units)                        978                  947    97%
          final size (Units)                        33                   30    91%
      Numeric final size (Units)                   124                  116    94%
      Numeric final size (MBytes)                  0.0                  0.0    94%
      peak memory usage (Units)                   1230                 1199    97%
      peak memory usage (MBytes)                   0.0                  0.0    97%
      numeric factorization flops          1.65000e+02          3.60000e+01    22%
      nz in L (incl diagonal)                       27                   16    59%
      nz in U (incl diagonal)                       37                   29    78%
      nz in L+U (incl diagonal)                     54                   35    65%
      largest front (# entries)                     49                   12    24%
      largest # rows in front                        7                    2    29%
      largest # columns in front                     7                    6    86%

      initial allocation ratio used:                 0.7
      # of forced updates due to frontal growth:     0
      nz in L (incl diagonal), if none dropped       16
      nz in U (incl diagonal), if none dropped       29
      number of small entries dropped                0
      nonzeros on diagonal of U:                     9
      min abs. value on diagonal of U:               0.00e+00
      max abs. value on diagonal of U:               4.68e-01
      estimate of reciprocal of condition number:    0.00e+00
      indices in compressed pattern:                 14
      numerical values stored in Numeric object:     34
      numeric factorization defragmentations:        1
      numeric factorization reallocations:           1
      costly numeric factorization reallocations:    1
      numeric factorization CPU time (sec):          0.02
      numeric factorization wallclock time (sec):    0.00
      numeric factorization mflops (CPU time):       0.00

      solve flops:                                   7.20000e+01
      iterative refinement steps taken:              0
      iterative refinement steps attempted:          0
      solve CPU time (sec):                          0.00
      solve wall clock time (sec):                   0.00

      total symbolic + numeric + solve flops:        1.08000e+02