m1_multiply_doc.html

张量分析工具

	    2.7470    5.0786    4.7737    2.7583
	    2.4439    4.5664    4.2765    2.4655
	    2.6095    4.8234    4.5165    2.6018
	Y(:,:,2,2) = 
	    1.4740    2.7639    2.5977    1.5023
	    2.8931    5.3300    4.9868    2.8703
	    2.5479    4.7381    4.4189    2.5385
	    2.7060    4.9891    4.6705    2.6895
	Y(:,:,3,2) = 
	    1.0365    1.9368    1.8275    1.0598
	    1.9305    3.6151    3.4161    1.9837
	    1.7213    3.2586    3.0731    1.7830
	    1.8406    3.4297    3.2279    1.8680
	Y(:,:,1,3) = 
	    1.3367    2.4889    2.3411    1.3540
	    2.5428    4.6564    4.3894    2.5403
	    2.2559    4.1595    3.9180    2.2671
	    2.4183    4.4405    4.1641    2.4005
	Y(:,:,2,3) = 
	    1.3373    2.4918    2.3410    1.3528
	    2.6485    4.8327    4.5351    2.6148
	    2.3258    4.2653    4.0063    2.3123
	    2.4723    4.5286    4.2431    2.4440
	Y(:,:,3,3) = 
	    0.9845    1.8150    1.7108    0.9905
	    1.8066    3.3375    3.1542    1.8302
	    1.6093    2.9879    2.8273    1.6426
	    1.7309    3.1876    2.9998    1.7345
</pre><pre class="codeinput">Y = ttm(X, {C,D}, [3 4]);    <span class="comment">%&lt;-- X times C in mode-3 &amp; D in mode-4</span>
Y = ttm(X, {A,B,C,D}, [3 4]) <span class="comment">%&lt;-- Same as above.</span>
</pre><pre class="codeoutput">Y is a tensor of size 5 x 3 x 3 x 3
	Y(:,:,1,1) = 
	    1.5822    1.3415    1.3949
	    1.2553    1.1977    1.2159
	    1.4244    1.6067    1.3862
	    1.2104    0.7492    1.5089
	    1.2932    1.3210    1.1591
	Y(:,:,2,1) = 
	    1.5112    1.0441    1.3712
	    1.1439    1.2984    1.2236
	    1.5205    1.6549    1.5110
	    1.3913    0.8995    1.5692
	    1.2733    1.4256    1.0567
	Y(:,:,3,1) = 
	    1.0920    1.0458    1.1802
	    0.9569    0.8498    0.8816
	    0.9992    1.0501    0.9638
	    0.8583    0.5944    1.0157
	    0.9676    1.0079    0.8150
	Y(:,:,1,2) = 
	    0.9474    0.8303    0.7452
	    0.8904    0.7530    0.6234
	    0.7403    0.9461    0.8350
	    0.8144    0.4605    0.7893
	    0.7803    0.6910    0.7280
	Y(:,:,2,2) = 
	    0.9542    0.7092    0.7569
	    0.8336    0.8635    0.6079
	    0.8297    0.9292    0.9154
	    0.8980    0.5883    0.7834
	    0.8064    0.7848    0.7567
	Y(:,:,3,2) = 
	    0.6673    0.6108    0.5736
	    0.6973    0.4752    0.4593
	    0.5460    0.6273    0.5737
	    0.5921    0.3631    0.5820
	    0.5514    0.4938    0.5138
	Y(:,:,1,3) = 
	    0.8673    0.7494    0.7182
	    0.7597    0.6751    0.6126
	    0.7228    0.8726    0.7625
	    0.7098    0.4168    0.7683
	    0.7120    0.6726    0.6529
	Y(:,:,2,3) = 
	    0.8539    0.6157    0.7186
	    0.7037    0.7561    0.6064
	    0.7919    0.8754    0.8338
	    0.7962    0.5187    0.7797
	    0.7207    0.7460    0.6432
	Y(:,:,3,3) = 
	    0.6056    0.5653    0.5783
	    0.5887    0.4485    0.4479
	    0.5208    0.5749    0.5266
	    0.5108    0.3295    0.5433
	    0.5160    0.4959    0.4601
</pre><pre class="codeinput">Y = ttm(X, {A,B,D}, [1 2 4]);   <span class="comment">%&lt;-- 3-way multiply.</span>
Y = ttm(X, {A,B,C,D}, [1 2 4]); <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, {A,B,D}, -3);        <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, {A,B,C,D}, -3)       <span class="comment">%&lt;-- Same as above.</span>
</pre><pre class="codeoutput">Y is a tensor of size 4 x 4 x 4 x 3
	Y(:,:,1,1) = 
	    1.0560    2.1553    2.0387    1.1911
	    2.4278    4.3898    4.2181    2.4754
	    2.1441    3.9274    3.7356    2.1771
	    2.2457    4.2196    3.9658    2.2936
	Y(:,:,2,1) = 
	    1.3324    2.2654    2.0972    1.1901
	    2.6489    4.6589    4.2674    2.4065
	    2.2210    3.9024    3.6500    2.0934
	    2.3874    4.1487    3.8458    2.1882
	Y(:,:,3,1) = 
	    1.5259    2.6902    2.5346    1.4625
	    2.4233    4.3577    4.1285    2.3958
	    2.1583    3.9709    3.7404    2.1642
	    2.4544    4.4132    4.1332    2.3769
	Y(:,:,4,1) = 
	    1.2470    2.3630    2.2327    1.2974
	    2.4531    4.5551    4.3305    2.5251
	    2.2263    4.0016    3.8576    2.2685
	    2.3016    4.2637    4.0530    2.3627
	Y(:,:,1,2) = 
	    0.6944    1.3989    1.3155    0.7645
	    1.4979    2.6951    2.5093    1.4364
	    1.3402    2.4234    2.2267    1.2614
	    1.4163    2.6071    2.4098    1.3735
	Y(:,:,2,2) = 
	    0.8962    1.5379    1.4173    0.8018
	    1.6626    3.0068    2.7746    1.5774
	    1.4220    2.6151    2.4249    1.3856
	    1.5214    2.7047    2.5241    1.4463
	Y(:,:,3,2) = 
	    0.8062    1.4427    1.3306    0.7556
	    1.3527    2.4504    2.3102    1.3362
	    1.2317    2.2839    2.1471    1.2410
	    1.3547    2.4308    2.2533    1.2850
	Y(:,:,4,2) = 
	    0.6588    1.3312    1.3007    0.7781
	    1.3637    2.6926    2.5765    1.5156
	    1.2072    2.3770    2.2702    1.3333
	    1.2520    2.4805    2.3848    1.4080
	Y(:,:,1,3) = 
	    0.6109    1.2374    1.1665    0.6794
	    1.3537    2.4408    2.3039    1.3333
	    1.2045    2.1900    2.0426    1.1717
	    1.2680    2.3547    2.1921    1.2574
	Y(:,:,2,3) = 
	    0.7812    1.3355    1.2330    0.6984
	    1.4915    2.6660    2.4525    1.3896
	    1.2650    2.2829    2.1244    1.2158
	    1.3562    2.3881    2.2224    1.2697
	Y(:,:,3,3) = 
	    0.7811    1.3880    1.2929    0.7398
	    1.2777    2.3069    2.1797    1.2626
	    1.1519    2.1284    2.0027    1.1581
	    1.2862    2.3100    2.1515    1.2317
	Y(:,:,4,3) = 
	    0.6383    1.2522    1.2053    0.7119
	    1.2905    2.4790    2.3654    1.3861
	    1.1555    2.1838    2.0942    1.2306
	    1.1966    2.2999    2.2001    1.2917
</pre><h2>Sparse tensor times matrix (ttm for sptensor)<a name="33"></a></h2>
         <p>It is also possible to multiply an sptensor times a matrix or series of matrices. The arguments are the same as for the dense
            case. The result may be dense or sparse, depending on its density.
         </p><pre class="codeinput">X = sptenrand([5 3 4 2],10);
Y = ttm(X, A, 1);         <span class="comment">%&lt;-- X times A in mode-1.</span>
Y = ttm(X, {A,B,C,D}, 1); <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, A', 1, <span class="string">'t'</span>)    <span class="comment">%&lt;-- Same as above</span>
</pre><pre class="codeoutput">Y is a sparse tensor of size 4 x 3 x 4 x 2 with 28 nonzeros
	(1,1,1,1)    0.1506
	(1,2,2,1)    0.0129
	(1,3,2,1)    0.0893
	(1,2,1,2)    0.4196
	(1,1,2,2)    0.0066
	(1,3,2,2)    0.0377
	(1,3,3,2)    0.4310
	(2,1,1,1)    0.7738
	(2,2,2,1)    0.3652
	(2,3,2,1)    0.4116
	(2,2,1,2)    0.2315
	(2,1,2,2)    0.1852
	(2,3,2,2)    0.2910
	(2,3,3,2)    0.7843
	(3,1,1,1)    0.8920
	(3,2,2,1)    0.2743
	(3,3,2,1)    0.3639
	(3,2,1,2)    0.3365
	(3,1,2,2)    0.1391
	(3,3,2,2)    0.2387
	(3,3,3,2)    0.9501
	(4,1,1,1)    0.5556
	(4,2,2,1)    0.3154
	(4,3,2,1)    0.1610
	(4,2,1,2)    0.4202
	(4,1,2,2)    0.1600
	(4,3,2,2)    0.1799
	(4,3,3,2)    0.7447
</pre><pre class="codeinput">norm(full(Y) - ttm(full(X),A, 1) ) <span class="comment">%&lt;-- Same as dense case.</span>
</pre><pre class="codeoutput">
ans =

     0

</pre><pre class="codeinput">Y = ttm(X, {A,B,C,D}, [1 2 3 4]); <span class="comment">%&lt;-- 4-way multiply.</span>
Y = ttm(X, {D,C,B,A}, [4 3 2 1]); <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, {A,B,C,D});            <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, {A',B',C',D'}, <span class="string">'t'</span>)    <span class="comment">%&lt;-- Same as above.</span>
</pre><pre class="codeoutput">Y is a tensor of size 4 x 4 x 3 x 3
	Y(:,:,1,1) = 
	    0.0890    0.1395    0.1470    0.0913
	    0.2458    0.5003    0.5113    0.3157
	    0.2497    0.5323    0.5461    0.3388
	    0.2063    0.4095    0.3985    0.2375
	Y(:,:,2,1) = 
	    0.0985    0.1757    0.1675    0.0976
	    0.3510    0.7241    0.7029    0.4190
	    0.3310    0.7403    0.7158    0.4273
	    0.2770    0.5858    0.5249    0.2942
	Y(:,:,3,1) = 
	    0.0419    0.0376    0.0617    0.0462
	    0.1291    0.1583    0.1999    0.1358
	    0.1288    0.1546    0.2055    0.1430
	    0.1050    0.1303    0.1583    0.1054
	Y(:,:,1,2) = 
	    0.0981    0.1224    0.1323    0.0821
	    0.1729    0.2599    0.3046    0.2013
	    0.1919    0.2865    0.3360    0.2219
	    0.1682    0.2465    0.2699    0.1710
	Y(:,:,2,2) = 
	    0.0973    0.1452    0.1299    0.0708
	    0.1977    0.3532    0.3625    0.2229
	    0.2021    0.3706    0.3744    0.2280
	    0.1835    0.3276    0.3052    0.1747
	Y(:,:,3,2) = 
	    0.0483    0.0371    0.0666    0.0510
	    0.1054    0.0994    0.1585    0.1176
	    0.1157    0.1031    0.1730    0.1305
	    0.0945    0.0895    0.1386    0.1018
	Y(:,:,1,3) = 
	    0.0719    0.0965    0.1034    0.0642
	    0.1480    0.2538    0.2795    0.1794
	    0.1587    0.2752    0.3040    0.1956
	    0.1361    0.2251    0.2346    0.1450
	Y(:,:,2,3) = 
	    0.0738    0.1169    0.1071    0.0599
	    0.1860    0.3555    0.3552    0.2152
	    0.1835    0.3684    0.3644    0.2199
	    0.1610    0.3092    0.2830    0.1605
	Y(:,:,3,3) = 
	    0.0350    0.0282    0.0491    0.0374
	    0.0853    0.0892    0.1296    0.0932
	    0.0905    0.0902    0.1384    0.1016
	    0.0739    0.0773    0.1094    0.0777
</pre><pre class="codeinput">Y = ttm(X, {C,D}, [3 4]);    <span class="comment">%&lt;-- X times C in mode-3 &amp; D in mode-4</span>
Y = ttm(X, {A,B,C,D}, [3 4]) <span class="comment">%&lt;-- Same as above.</span>
</pre><pre class="codeoutput">Y is a tensor of size 5 x 3 x 3 x 3
	Y(:,:,1,1) = 
	         0    0.1004    0.0998
	    0.4075         0    0.1363
	    0.0221    0.1332    0.0167
	         0         0    0.3307
	         0         0         0
	Y(:,:,2,1) = 
	         0    0.1323    0.0395
	    0.5371         0    0.0540
	    0.0427    0.2578    0.0322
	         0         0    0.6257
	         0         0         0
	Y(:,:,3,1) = 
	         0    0.0154    0.0943
	    0.0626         0    0.1289
	    0.0123    0.0742    0.0093
	         0         0    0.1878
	         0         0         0
	Y(:,:,1,2) = 
	         0    0.1342    0.1334
	    0.1478         0    0.1823
	    0.0295    0.0483    0.0223
	         0         0    0.1626
	         0         0         0
	Y(:,:,2,2) = 
	         0    0.1769    0.0528
	    0.1948         0    0.0722
	    0.0571    0.0935    0.0431
	         0         0    0.2955
	         0         0         0
	Y(:,:,3,2) = 
	         0    0.0206    0.1262
	    0.0227         0    0.1724
	    0.0164    0.0269    0.0124
	         0         0    0.0954
	         0         0         0
	Y(:,:,1,3) = 
	         0    0.0933    0.0927
	    0.1737         0    0.1267
	    0.0205    0.0568    0.0155
	         0         0    0.1630
	         0         0         0
	Y(:,:,2,3) = 
	         0    0.1229    0.0367
	    0.2290         0    0.0502
	    0.0397    0.1099    0.0300
	         0         0    0.3022
	         0         0         0
	Y(:,:,3,3) = 
	         0    0.0143    0.0877
	    0.0267         0    0.1198
	    0.0114    0.0316    0.0086
	         0         0    0.0942
	         0         0         0
</pre><pre class="codeinput">Y = ttm(X, {A,B,D}, [1 2 4]);   <span class="comment">%&lt;-- 3-way multiply.</span>
Y = ttm(X, {A,B,C,D}, [1 2 4]); <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, {A,B,D}, -3);        <span class="comment">%&lt;-- Same as above.</span>
Y = ttm(X, {A,B,C,D}, -3)       <span class="comment">%&lt;-- Same as above.</span>
</pre><pre class="codeoutput">Y is a tensor of size 4 x 4 x 4 x 3
	Y(:,:,1,1) = 
	    0.1010    0.2463    0.1918    0.0950
	    0.1515    0.7021    0.6066    0.3415
	    0.1879    0.8297    0.7130    0.3989
	    0.1573    0.5785    0.4856    0.2645
	Y(:,:,2,1) = 
	    0.0416    0.0313    0.0563    0.0431
	    0.3903    0.4667    0.4983    0.3058
	    0.3136    0.3615    0.4026    0.2533
	    0.2647    0.3646    0.3297    0.1801
	Y(:,:,3,1) = 
	    0.0468    0.0249    0.0651    0.0543
	    0.0851    0.0453    0.1184    0.0988
	    0.1031    0.0549    0.1434    0.1196
	    0.0808    0.0430    0.1124    0.0938
	Y(:,:,4,1) = 
	     0     0     0     0
	     0     0     0     0
	     0     0     0     0
	     0     0     0     0
	Y(:,:,1,2) = 
	    0.1147    0.2091    0.1501    0.0656
	    0.0980    0.3207    0.2645    0.1410
	    0.1308    0.3970    0.3232    0.1696
	    0.1353    0.3298    0.2568    0.1271
	Y(:,:,2,2) = 
	    0.0194    0.0152    0.0275    0.0211
	    0.1805    0.2340    0.2663    0.1713
	    0.1451    0.1809    0.2133    0.1397
	    0.1221    0.1841    0.1830    0.1087
	Y(:,:,3,2) = 
	    0.0626    0.0333    0.0870    0.0726
	    0.1138    0.0606    0.1584    0.1321
	    0.1379    0.0735    0.1919    0.1600
	    0.1081    0.0576    0.1504    0.1254
	Y(:,:,4,2) = 
	     0     0     0     0
	     0     0     0     0
	     0     0     0     0
	     0     0     0     0
	Y(:,:,1,3) = 
	    0.0834    0.1668    0.1233    0.0566
	    0.0868    0.3335    0.2816    0.1545
	    0.1125    0.4033    0.3373    0.1830
	    0.1074    0.3086    0.2487    0.1289
	Y(:,:,2,3) = 
	    0.0200    0.0153    0.0277    0.0212
	    0.1865    0.2324    0.2566    0.1616
	    0.1499    0.1798    0.2064    0.1327
	    0.1263    0.1822    0.1733    0.0992
	Y(:,:,3,3) = 
	    0.0435    0.0232    0.0605    0.0504
	    0.0791    0.0421    0.1100    0.0918
	    0.0958    0.0510    0.1333    0.1112
	    0.0751    0.0400    0.1045    0.0872
	Y(:,:,4,3) = 
	     0     0     0     0
	     0     0     0     0
	     0     0     0     0
	     0     0     0     0
</pre><p>The result may be dense or sparse.</p><pre class="codeinput">X = sptenrand([5 3 4],1);
Y = ttm(X, A, 1) <span class="comment">%&lt;-- Sparse result.</span>
</pre><pre class="codeoutput">Y is a sparse tensor of size 4 x 3 x 4 with 4 nonzeros
	(1,3,2)    0.0232
	(2,3,2)    0.1067
	(3,3,2)    0.0943
	(4,3,2)    0.0417
</pre><pre class="codeinput">X = sptenrand([5 3 4],50);
Y = ttm(X, A, 1) <span class="comment">%&lt;-- Dense result.</span>
</pre><pre class="codeoutput">Y is a tensor of size 4 x 3 x 4
	Y(:,:,1) = 
	    0.4159    0.5631    0.0406
	    0.9765    1.4239    0.2088
	    0.9029    1.2234    0.2406
	    0.8744    1.2606    0.1499
	Y(:,:,2) = 
	    1.0127    0.5477         0
	    1.4153    1.1141         0
	    1.1989    1.0105         0
	    1.6381    0.9835         0
	Y(:,:,3) = 
	    0.5923    0.6934    0.9184
	    0.8260    0.9650    0.8641
	    0.4955    0.7236    0.5323
	    0.8762    0.9604    1.0351
	Y(:,:,4) = 
	    1.2906    0.7036    0.4899
	    1.2638    0.8427    1.1469
	    1.1332    0.8936    1.0464
	    1.5305    0.9649    0.9588