Interesting. FFT for this task, who would have thought a few days ago. Looks like it runs in almost constant time, regardless of kernel size; and of course slower than special solutions. I had to add a couple of checks to accommodate for (unimportant) corner test case of submatrix with dims (4,3) and example matrix.

sub sms_WxH_PDL_fftconvolve ( $m, $w, $h ) {
    my ( $W, $H ) = $m-> dims;

    my $small = ones( $w%2?$w:$w+1, $h%2?$h:$h+1 );
    $small-> slice( -1 )    .= 0 unless $w%2; # zero row and/or column
+ for even kernels
    $small-> slice( [],-1 ) .= 0 unless $h%2;

    my $result = $m-> copy;
    $result = append( $result, 0 ) if $W == $w and not $W % 2;
    $result = append( $result-> transpose, 0 )-> transpose 
                                   if $H == $h and not $H % 2;

    my $kernel = kernctr( $result, $small ); #full kernel
    $result-> fftconvolve( $kernel );
    
    $result = $result-> slice( '0:-2','' ) if $W == $w and not $W % 2;
    $result = $result-> slice( '','0:-2' ) if $H == $h and not $H % 2;

    return $result-> slice(
        [floor(($w-1)/2),floor(-($w+1)/2)],
        [floor(($h-1)/2),floor(-($h+1)/2)]
    )
}

__END__

    Time (s) vs. N (NxN submatrix, PDL: Double D [1500,1500] matrix)

      +-----------------------------------------------------------+
      |    +      +      +      +       +      +      +      +    |
      |     A                                                     |
      |                                                           |
      |                                                           |
    2 |-+                                                       +-|
      |    B   B                        B  B      B               |
      |    BB         B      B      B          B         B   B    |
      |     B     B      B      B                     B           |
      |                         C   C                             |
      |                      C          C                         |
  1.5 |-+   A                              C                    +-|
      |                  C                                        |
      |                                        C                  |
      |                                                           |
      |               C                                           |
    1 |-+                                         C             +-|
      |     A                                                     |
      |           C                                   C           |
      |                                                           |
      |                                                           |
      |                                                  C        |
  0.5 |-+  A                                                    +-|
      |        C                                                  |
      |                                                      C    |
      |    A                                                      |
      |     D  D                                                  |
      |    CC     D   D  D   D  D   D   D  D   D  D   D  D        |
    0 |-+                                                    D  +-|
      |    +      +      +      +       +      +      +      +    |
      +-----------------------------------------------------------+
           0     200    400    600     800    1000   1200   1400
                         sms_WxH_PDL_conv2d    A
                    sms_WxH_PDL_fftconvolve    B
                           sms_WxH_PDL_lags    C
                        sms_WxH_PDL_sliding    D


  +------+-------+-------+-------+-------+
  | N    | A     | B     | C     | D     |
  +------+-------+-------+-------+-------+
  | 2    | 0.023 | 1.792 | 0.021 |       |
  | 4    | 0.065 | 1.854 | 0.036 |       |
  | 8    | 0.237 | 1.896 | 0.031 |       |
  | 12   | 0.534 | 1.865 | 0.049 |       |
  | 16   | 0.930 | 1.740 | 0.039 |       |
  | 20   | 1.432 | 1.812 | 0.057 |       |
  | 25   | 2.224 | 1.716 | 0.081 | 0.099 |
  | 100  |       | 1.899 | 0.380 | 0.107 |
  | 200  |       | 1.779 | 0.826 | 0.075 |
  | 300  |       | 1.836 | 1.143 | 0.089 |
  | 400  |       | 1.760 | 1.393 | 0.065 |
  | 500  |       | 1.826 | 1.560 | 0.047 |
  | 600  |       | 1.740 | 1.659 | 0.036 |
  | 700  |       | 1.818 | 1.620 | 0.049 |
  | 800  |       | 1.919 | 1.589 | 0.031 |
  | 900  |       | 1.917 | 1.484 | 0.026 |
  | 1000 |       | 1.789 | 1.260 | 0.008 |
  | 1100 |       | 1.880 | 1.068 | 0.010 |
  | 1200 |       | 1.745 | 0.823 | 0.010 |
  | 1300 |       | 1.841 | 0.562 | 0.008 |
  | 1400 |       | 1.841 | 0.273 | 0.003 |
  +------+-------+-------+-------+-------+

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In reply to Re^4: Fast sliding submatrix sums with PDL (inspired by PWC 248 task 2) by Anonymous Monk
in thread Fast sliding submatrix sums with PDL (inspired by PWC 248 task 2) by Anonymous Monk

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