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How to suppress complex numbers in the calculation of the square root of a positive semi-definite matrix using scipy.linalg.sqrtm()

See original GitHub issue

Say I have a matrix A (this matrix is quite large so I did not specify this here), and applying scipy.linalg.sqrtm(A), I get the following output matrix which contains complex numbers:

array([[ 64.06444692+8.96831017e-44j,  -6.28785787+1.57772181e-30j,
         -9.15586255+3.94430453e-31j,  -9.00198187+1.97215226e-31j,
         -7.71376342+7.88860905e-31j,  -6.28785787-1.97215226e-30j,
          2.97380679+0.00000000e+00j,   1.29893642+7.88860905e-31j,
          1.31682879+1.57772181e-30j,  -0.351662  +7.88860905e-31j,
         -9.15586255-6.90253292e-31j,   1.29893642+1.97215226e-31j,
          2.36636534+4.48415509e-44j,  -0.57657211+9.86076132e-31j,
          0.39917092-2.95822839e-31j,  -9.00198187-3.94430453e-31j,
          1.31682879-1.57772181e-30j,  -0.57657211-1.97215226e-31j,
          2.36648045+4.48415509e-44j,   0.28162   +1.97215226e-31j,
         -7.71376342-7.88860905e-31j,  -0.351662  +0.00000000e+00j,
          0.39917092+6.90253292e-31j,   0.28162   -3.94430453e-31j,
          1.21720917+3.58732407e-43j],
       [ -6.28785787+0.00000000e+00j,  37.43772093+0.00000000e+00j,
         -4.54039284+6.61744490e-24j,  -4.70636743+3.30872245e-24j,
         -2.75237216+6.61744490e-24j,  37.43772062-2.64697796e-23j,
         -5.74532523-2.25694915e-36j,  -3.84385982-1.32348898e-23j,
         -4.65882698+3.30872245e-24j,  -2.99673941+2.64697796e-23j,
         -4.54039267+0.00000000e+00j,  -3.84386008+1.32348898e-23j,
         -0.51143448+3.00926554e-36j,  -6.96326972+3.97046694e-23j,
          2.14326501-3.97046694e-23j,  -4.7063676 +0.00000000e+00j,
         -4.65882709-3.30872245e-24j,  -6.96326948+6.61744490e-24j,
          0.72851489+1.69271187e-36j,   2.15291748+0.00000000e+00j,
         -2.75237195-1.32348898e-23j,  -2.99674005-2.64697796e-23j,
          2.14326499+2.64697796e-23j,   2.15291766+0.00000000e+00j,
          0.2575331 +0.00000000e+00j],
       [ -9.15586255-3.76158192e-37j,  -4.54039278+8.27180613e-24j,
         32.72911898-4.96308368e-24j,  -1.85021394-1.65436123e-24j,
         -2.53702773+9.92616735e-24j,  -4.54039273-1.48892510e-23j,
          1.83872342-1.12847458e-36j,  -1.86092192-6.61744490e-24j,
         -4.29572397+9.92616735e-24j,   1.09027491-2.64697796e-23j,
         32.72911818+0.00000000e+00j,  -1.86092174+6.61744490e-24j,
         -9.24894513-9.40395481e-38j,  -4.93062942+3.30872245e-24j,
         -2.40960071+2.31610572e-23j,  -1.85021376+0.00000000e+00j,
         -4.29572395-6.61744490e-24j,  -4.93062949+0.00000000e+00j,
          1.40879764+6.58276836e-37j,   1.34936369+1.98523347e-23j,
         -2.53702818-5.79026429e-24j,   1.090275  +1.32348898e-23j,
         -2.40960075-1.98523347e-23j,   1.34936389-1.65436123e-23j,
          0.47563635+3.00926554e-36j],
       [ -9.00198187+5.17217514e-36j,  -4.70636737+3.30872245e-23j,
         -1.85021382+1.65436123e-23j,  32.3679699 -3.30872245e-24j,
         -2.62837621+1.65436123e-23j,  -4.70636767-3.30872245e-23j,
          1.64638076-7.52316385e-37j,  -4.2159266 -2.97785021e-23j,
         -0.86792129+6.61744490e-24j,   0.99791227-5.29395592e-23j,
         -1.85021388-1.15805286e-23j,  -4.21592618+2.81241408e-23j,
          1.17619748+6.01853108e-36j,  -3.93042699-6.61744490e-24j,
          1.17237038+6.61744490e-24j,  32.36796885+6.61744490e-24j,
         -0.86792107-6.61744490e-24j,  -3.93042699+6.61744490e-24j,
        -10.37559738+1.12847458e-36j,  -2.60535084-5.62482817e-23j,
         -2.62837615-1.65436123e-23j,   0.99791236+4.63221143e-23j,
          1.17237044-6.61744490e-24j,  -2.60535074+5.62482817e-23j,
          0.59574761-3.00926554e-36j],
       [ -7.71376342+4.51389831e-36j,  -2.75237237+1.98523347e-23j,
         -2.53702781+3.30872245e-24j,  -2.62837605+3.30872245e-24j,
          5.69625179+1.32348898e-23j,  -2.75237175-1.98523347e-23j,
         -0.56750495+1.12847458e-36j,   1.03933818-1.32348898e-23j,
          1.0435955 -6.61744490e-24j,   0.3804354 -1.32348898e-23j,
         -2.5370281 -3.30872245e-24j,   1.03933817+1.32348898e-23j,
          0.42008941+3.00926554e-36j,   2.3668576 -6.61744490e-24j,
         -0.24619723-5.29395592e-23j,  -2.62837631+0.00000000e+00j,
          1.04359542+0.00000000e+00j,   2.3668577 +8.27180613e-24j,
          0.34120424+3.76158192e-37j,  -0.2668792 -5.29395592e-23j,
          5.6962513 -6.61744490e-24j,   0.38043513-6.61744490e-24j,
         -0.24619695+3.97046694e-23j,  -0.2668791 +4.30133919e-23j,
         -1.24019775+0.00000000e+00j],
       [ -6.28785787+0.00000000e+00j,  37.43772062-2.64697796e-23j,
         -4.54039267-1.32348898e-23j,  -4.7063676 +0.00000000e+00j,
         -2.75237195-3.30872245e-24j,  37.43772093+2.64697796e-23j,
         -5.74532523+7.52316385e-37j,  -3.84386008+0.00000000e+00j,
         -4.65882709+0.00000000e+00j,  -2.99674005-2.64697796e-23j,
         -4.54039284+1.98523347e-23j,  -3.84385982+0.00000000e+00j,
         -0.51143448-3.00926554e-36j,  -6.96326948-2.64697796e-23j,
          2.14326499+3.97046694e-23j,  -4.70636743-6.61744490e-24j,
         -4.65882698+0.00000000e+00j,  -6.96326972+0.00000000e+00j,
          0.72851489-1.78675141e-36j,   2.15291766+0.00000000e+00j,
         -2.75237216+3.30872245e-24j,  -2.99673941+1.32348898e-23j,
          2.14326501-3.30872245e-23j,   2.15291748-1.32348898e-23j,
          0.2575331 +0.00000000e+00j],
       [  2.97380679+2.24207754e-44j,  -5.74532523+0.00000000e+00j,
          1.83872342-1.57772181e-30j,   1.64638076-3.69778549e-31j,
         -0.56750495+1.57772181e-30j,  -5.74532523+7.88860905e-31j,
         66.4932094 +0.00000000e+00j, -10.70816264+2.56379794e-30j,
        -11.08111069+1.97215226e-31j,  -7.43474479-1.97215226e-31j,
          1.83872342+7.88860905e-31j, -10.70816264-1.57772181e-30j,
          1.86080059+2.69049305e-43j,   0.19541758+7.88860905e-31j,
          0.56304712+1.57772181e-30j,   1.64638076+3.20474743e-31j,
        -11.08111069+1.97215226e-31j,   0.19541758+0.00000000e+00j,
          1.96616419+4.48415509e-44j,   0.54752066-1.18329136e-30j,
         -0.56750495-1.57772181e-30j,  -7.43474479-3.94430453e-31j,
          0.56304712-1.57772181e-30j,   0.54752066+1.08468374e-30j,
          1.07252937+5.38098610e-43j],
       [  1.29893642+9.78011300e-36j,  -3.84386014+5.29395592e-23j,
         -1.86092176+9.92616735e-24j,  -4.21592632+1.32348898e-23j,
          1.03933803+6.61744490e-24j,  -3.84385976-6.61744490e-23j,
        -10.70816264+1.88079096e-36j,  31.76054462-5.29395592e-23j,
         -0.81206901+1.98523347e-23j,  -2.42506017-7.27918939e-23j,
         -1.8609219 -2.64697796e-23j,  31.76054419+2.64697796e-23j,
         -9.99217824+6.01853108e-36j,  -4.18589448-4.30133919e-23j,
         -2.25738823+0.00000000e+00j,  -4.21592646+0.00000000e+00j,
         -0.81206922-6.61744490e-24j,  -4.18589417+4.63221143e-23j,
          1.53827682+1.88079096e-37j,   1.02240603-1.45583788e-22j,
          1.03933832-6.61744490e-24j,  -2.42506066+7.27918939e-23j,
         -2.2573882 -1.32348898e-23j,   1.02240648+1.45583788e-22j,
          0.4993346 -1.20370622e-35j],
       [  1.31682879+9.02779661e-36j,  -4.65882716+3.30872245e-23j,
         -4.29572396+1.32348898e-23j,  -0.86792122+7.44462551e-24j,
          1.04359537+2.64697796e-23j,  -4.65882691-5.29395592e-23j,
        -11.08111069+3.00926554e-36j,  -0.81206926-5.29395592e-23j,
         32.00910946+1.32348898e-23j,  -2.29180333-6.61744490e-23j,
         -4.29572396-1.98523347e-23j,  -0.81206897+7.94093388e-23j,
          1.42042175+4.51389831e-36j,  -3.87247572-2.97785021e-23j,
          0.971801  -1.98523347e-23j,  -0.86792115-2.48154184e-24j,
         32.00910859-2.64697796e-23j,  -3.87247562+3.63959470e-23j,
        -10.69471254-9.40395481e-38j,  -2.38335511-7.94093388e-23j,
          1.04359555-1.32348898e-23j,  -2.29180342+7.27918939e-23j,
          0.97180114+0.00000000e+00j,  -2.38335456+1.19114008e-22j,
          0.51309298-3.00926554e-36j],
       [ -0.351662  -6.77084746e-36j,  -2.99673978-2.64697796e-23j,
          1.09027494-2.64697796e-23j,   0.99791231-3.30872245e-24j,
          0.38043531+6.61744490e-24j,  -2.99673969+2.64697796e-23j,
         -7.43474479-1.78675141e-36j,  -2.42506037+2.31610572e-23j,
         -2.29180337-4.96308368e-24j,   5.48456743+5.29395592e-23j,
          1.09027497+1.98523347e-23j,  -2.42506046-1.15805286e-23j,
          0.66133065-4.51389831e-36j,   2.08403436+3.63959470e-23j,
         -0.34767484-4.30133919e-23j,   0.99791231+4.96308368e-24j,
         -2.29180338+5.79026429e-24j,   2.08403451-1.32348898e-23j,
          0.53327358+1.03443503e-36j,  -0.33110204+7.23783036e-23j,
          0.38043522-6.61744490e-24j,   5.48456708-5.29395592e-23j,
         -0.34767495+3.97046694e-23j,  -0.33110231-7.63074115e-23j,
         -1.13338852+1.20370622e-35j],
       [ -9.15586255+1.50463277e-36j,  -4.54039273-4.13590306e-24j,
         32.72911818+4.96308368e-24j,  -1.85021376+1.65436123e-24j,
         -2.53702818-1.32348898e-23j,  -4.54039278-2.48154184e-24j,
          1.83872342+1.50463277e-36j,  -1.86092174+0.00000000e+00j,
         -4.29572395+0.00000000e+00j,   1.090275  +1.32348898e-23j,
         32.72911898-6.61744490e-24j,  -1.86092192+9.92616735e-24j,
         -9.24894513-5.64237288e-37j,  -4.93062949-9.92616735e-24j,
         -2.40960075-1.32348898e-23j,  -1.85021394-1.65436123e-24j,
         -4.29572397+0.00000000e+00j,  -4.93062942+6.61744490e-24j,
          1.40879764-3.76158192e-37j,   1.34936389-3.97046694e-23j,
         -2.53702773-1.65436123e-24j,   1.09027491-2.64697796e-23j,
         -2.40960071+6.61744490e-24j,   1.34936369+3.97046694e-23j,
          0.47563635-1.50463277e-35j],
       [  1.29893642-1.05324294e-35j,  -3.84385976-5.95570041e-23j,
         -1.8609219 -2.64697796e-23j,  -4.21592646-6.61744490e-24j,
          1.03933832+2.31610572e-23j,  -3.84386014+6.61744490e-23j,
        -10.70816264-3.38542373e-36j,  31.76054419+5.29395592e-23j,
         -0.81206922+1.98523347e-23j,  -2.42506066+5.95570041e-23j,
         -1.86092176+2.31610572e-23j,  31.76054462-7.94093388e-23j,
         -9.99217824-6.77084746e-36j,  -4.18589417+3.30872245e-23j,
         -2.2573882 +3.97046694e-23j,  -4.21592632+6.61744490e-24j,
         -0.81206901-1.32348898e-23j,  -4.18589448-4.63221143e-23j,
          1.53827682-1.12847458e-36j,   1.02240648+1.05879118e-22j,
          1.03933803+9.92616735e-24j,  -2.42506017-5.29395592e-23j,
         -2.25738823-1.32348898e-23j,   1.02240603-1.05879118e-22j,
          0.4993346 +0.00000000e+00j],
       [  2.36636534+8.96831017e-44j,  -0.51143448-1.57772181e-30j,
         -9.24894513-8.87468518e-31j,   1.17619748+3.94430453e-31j,
          0.42008941-7.95023881e-31j,  -0.51143448+2.36658272e-30j,
          1.86080059+1.79366203e-43j,  -9.99217824+1.77493704e-30j,
          1.42042175-7.88860905e-31j,   0.66133065+0.00000000e+00j,
         -9.24894513+5.91645679e-31j,  -9.99217824-1.18329136e-30j,
         64.80119411-1.79366203e-43j,  -5.84612899-1.08468374e-30j,
         -7.81047186+0.00000000e+00j,   1.17619748-5.91645679e-31j,
          1.42042175+0.00000000e+00j,  -5.84612899+1.77493704e-30j,
          3.7271156 -2.24207754e-44j,  -0.60817689+0.00000000e+00j,
          0.42008941+4.25245332e-31j,   0.66133065+0.00000000e+00j,
         -7.81047186+7.88860905e-31j,  -0.60817689+1.97215226e-31j,
          1.20175111+6.27781712e-43j],
       [ -0.57657211+1.50463277e-36j,  -6.96326969+2.64697796e-23j,
         -4.93062959+1.65436123e-24j,  -3.93042692-6.61744490e-24j,
          2.36685766+1.32348898e-23j,  -6.96326951-1.32348898e-23j,
          0.19541758+0.00000000e+00j,  -4.18589425+1.32348898e-23j,
         -3.87247569-1.98523347e-23j,   2.08403463-3.97046694e-23j,
         -4.93062931+3.30872245e-24j,  -4.1858944 -1.65436123e-23j,
         -5.84612899+3.00926554e-36j,  37.77920832+3.30872245e-24j,
         -3.01763347+0.00000000e+00j,  -3.93042706+9.92616735e-24j,
         -3.87247565+1.65436123e-23j,  37.77920777-1.65436123e-24j,
         -7.23213763+1.50463277e-36j,  -3.04026415+6.61744490e-24j,
          2.36685764-1.32348898e-23j,   2.08403425+3.97046694e-23j,
         -3.01763353+6.61744490e-24j,  -3.04026397-6.61744490e-24j,
          0.28030407+1.20370622e-35j],
       [  0.39917092+2.44502825e-36j,   2.14326495+1.07533480e-23j,
         -2.40960073+6.61744490e-24j,   1.17237044+1.34416850e-24j,
         -0.24619708+0.00000000e+00j,   2.14326504-8.27180613e-24j,
          0.56304712+1.59867232e-36j,  -2.25738823-1.65436123e-23j,
          0.97180111-6.61744490e-24j,  -0.34767488+9.92616735e-24j,
         -2.40960073+3.30872245e-24j,  -2.25738819+6.61744490e-24j,
         -7.81047186-4.70197740e-37j,  -3.01763354-4.96308368e-24j,
          5.63841366-1.32348898e-23j,   1.17237037-3.30872245e-24j,
          0.97180102+0.00000000e+00j,  -3.01763345+6.61744490e-24j,
         -0.46208872+1.88079096e-37j,   0.35404579-3.30872245e-23j,
         -0.24619711+0.00000000e+00j,  -0.34767491-6.61744490e-24j,
          5.6384134 +1.98523347e-23j,   0.35404595+4.30133919e-23j,
         -1.1998646 -3.00926554e-36j],
       [ -9.00198187-6.58276836e-36j,  -4.70636767-3.97046694e-23j,
         -1.85021388-2.15066959e-23j,  32.36796885+0.00000000e+00j,
         -2.62837615-3.30872245e-24j,  -4.70636737+2.64697796e-23j,
          1.64638076+7.52316385e-37j,  -4.21592618+2.15066959e-23j,
         -0.86792107-1.98523347e-23j,   0.99791236+6.61744490e-23j,
         -1.85021382+2.64697796e-23j,  -4.2159266 -2.64697796e-23j,
          1.17619748-6.01853108e-36j,  -3.93042699+1.98523347e-23j,
          1.17237044-2.31610572e-23j,  32.3679699 +0.00000000e+00j,
         -0.86792129-1.32348898e-23j,  -3.93042699-1.65436123e-23j,
        -10.37559738+3.76158192e-37j,  -2.60535074+6.61744490e-23j,
         -2.62837621+3.30872245e-24j,   0.99791227-6.61744490e-23j,
          1.17237038+1.65436123e-23j,  -2.60535084-6.61744490e-23j,
          0.59574761+1.50463277e-36j],
       [  1.31682879-9.78011300e-36j,  -4.65882691-5.95570041e-23j,
         -4.29572396-6.61744490e-24j,  -0.86792115+4.96308368e-24j,
          1.04359555+0.00000000e+00j,  -4.65882716+4.63221143e-23j,
        -11.08111069-1.78675141e-36j,  -0.81206897+7.94093388e-23j,
         32.00910859+6.61744490e-24j,  -2.29180342+6.61744490e-23j,
         -4.29572396+2.64697796e-23j,  -0.81206926-6.61744490e-23j,
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          0.35404589+0.00000000e+00j,   5.62390689+6.28657266e-23j,
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       [ -7.71376342-6.01853108e-36j,  -2.75237175-1.32348898e-23j,
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          1.0435955 +0.00000000e+00j,   2.3668576 -1.15805286e-23j,
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         -0.24619723-3.97046694e-23j,  -0.2668792 -8.27180613e-23j,
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       [ -0.351662  +6.77084746e-36j,  -2.99673969+3.97046694e-23j,
          1.09027497+1.32348898e-23j,   0.99791231+4.13590306e-24j,
          0.38043522-3.30872245e-24j,  -2.99673978-2.64697796e-23j,
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         -1.24019775+0.00000000e+00j,   0.2575331 -5.52202634e-30j,
          1.07252937-3.58732407e-43j,   0.4993346 +0.00000000e+00j,
          0.51309298-1.26217745e-29j,  -1.13338852-6.31088724e-30j,
          0.47563635-1.57772181e-30j,   0.4993346 -6.31088724e-30j,
          1.20175111+8.96831017e-43j,   0.28030407+9.46633086e-30j,
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          1.10890059+0.00000000e+00j,  -1.18245232-3.15544362e-30j,
         -1.24019775+9.46633086e-30j,  -1.13338852+7.88860905e-30j,
         -1.1998646 -1.89326617e-29j,  -1.18245232+0.00000000e+00j,
          1.80359473-3.22859166e-42j]])

How do I suppress complex numbers within the calculation of scipy.linalg.sqrtm() so that the output matrix does not contain any complex numbers?

I did some googling around and found the following Github PR from 6 years ago which suggest to include function features like funm_psd and structure="psd" to evaluate general matrix functions of positive semi-definite matrices but I can’t seem to find these functions features anywhere now.

https://github.com/scipy/scipy/pull/3556

Any help would really be appreciated!

Edit: I know I can use np.real() on the output matrix to get just the real part. My concern is including complex numbers during the calculation of the square root might affect the values in the output matrix, rather than just removing the complex numbers in the output matrix. Unless of course if including or excluding the complex numbers during the calculation of the square root won’t affect much on the values of the output matrix. Would anyone know if this is the case?

I should also clarify that why I think the complex numbers are appearing is because of numerical precision/rounding issues with the implementation of the scipy.linalg.sqrtm() function (for eg. see here https://math.stackexchange.com/questions/909814/how-to-avoid-complex-value-for-square-root-of-a-symmetric-matrix). For eg. values which are mathematically suppose to be zeros, have values like -7.52206733e-20 + 7.73635801e-19j where you can see has a complex part but the real part is very very close to zero.

Issue Analytics

  • State:closed
  • Created 3 years ago
  • Comments:5 (5 by maintainers)

github_iconTop GitHub Comments

1reaction
ilayncommented, Oct 19, 2020

Yes indeed, you can check afterwards your result by forming the original matrix from the square root to verify.

1reaction
ilayncommented, Oct 19, 2020

These imaginary parts are, though annoying, numerically zero. Note that machine precision is around 2e-16. Hence it’s just noise. In your last sentence that entry is also zero. This is quite typical to have when the matrices grow large for many routines and not just sqrtm.

So you can actually clean up the noise afterwards which is what #3556 with a very numerically expensive eigh call. Or instead, you can check (np.abs(a.imag) < np.spacing(1.)).all() and then take the real part.

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