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Efficient way to sum over an index repeated more than twice

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5

2

$begingroup$

I'm looking for an efficient way of computing sums of the type
$$D_jk = sum_i=1^n A_ij B_ik C_i$$
for large numerical matrices $A$, $B$ and $C$.

Here is a slow solution:

DD = Table[Sum[AA[[i, j]] BB[[i, k]] CC[[i]], i, 1, n], j, 1, m, k, 1, m]

What's a fast way of doing it?

linear-algebra

share|improve this question

edited Jun 16 at 5:31

arne

asked Jun 16 at 3:12

arnearne

12855 bronze badges

$endgroup$

add a comment
 | 

5

2

$begingroup$

I'm looking for an efficient way of computing sums of the type
$$D_jk = sum_i=1^n A_ij B_ik C_i$$
for large numerical matrices $A$, $B$ and $C$.

Here is a slow solution:

DD = Table[Sum[AA[[i, j]] BB[[i, k]] CC[[i]], i, 1, n], j, 1, m, k, 1, m]

What's a fast way of doing it?

linear-algebra

share|improve this question

edited Jun 16 at 5:31

arne

asked Jun 16 at 3:12

arnearne

12855 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

I'm looking for an efficient way of computing sums of the type
$$D_jk = sum_i=1^n A_ij B_ik C_i$$
for large numerical matrices $A$, $B$ and $C$.

Here is a slow solution:

DD = Table[Sum[AA[[i, j]] BB[[i, k]] CC[[i]], i, 1, n], j, 1, m, k, 1, m]

What's a fast way of doing it?

linear-algebra

share|improve this question

edited Jun 16 at 5:31

arne

asked Jun 16 at 3:12

arnearne

12855 bronze badges

$endgroup$

DD = Table[Sum[AA[[i, j]] BB[[i, k]] CC[[i]], i, 1, n], j, 1, m, k, 1, m]

share|improve this question

edited Jun 16 at 5:31

arne

asked Jun 16 at 3:12

arnearne

12855 bronze badges

share|improve this question

edited Jun 16 at 5:31

arne

asked Jun 16 at 3:12

arnearne

12855 bronze badges

asked Jun 16 at 3:12

arnearne

12855 bronze badges

asked Jun 16 at 3:12

arnearne

12855 bronze badges

1 Answer
1

active

oldest

votes

8

$begingroup$

n = 4;
m = 3;
aa = Array[a, n, m];
bb = Array[b, n, m];
cc = Array[c, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m];

You can Transpose aa and Dot it with bb cc:

Transpose[aa].(bb cc) == dd

True

Alternatively, you can Transpose aa cc and Dot it with bb:

Transpose[aa cc].bb ==dd

True

Comparison with Table/Sum combination:

SeedRandom[1]
n = 100;
m = 50;
aa = RandomReal[1, n, m];
bb = RandomReal[1, n, m];
cc = RandomReal[1, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m]; //
 RepeatedTiming // First

0.417

ee1 = Transpose[aa].( bb cc) ; // RepeatedTiming // First

0.000051

ee2 = Transpose[aa cc].bb ; // RepeatedTiming // First

0.000045

dd == ee1 == ee2

True

share|improve this answer

edited Jun 16 at 4:00

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges

$endgroup$

add a comment
 | 

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8

$begingroup$

n = 4;
m = 3;
aa = Array[a, n, m];
bb = Array[b, n, m];
cc = Array[c, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m];

You can Transpose aa and Dot it with bb cc:

Transpose[aa].(bb cc) == dd

True

Alternatively, you can Transpose aa cc and Dot it with bb:

Transpose[aa cc].bb ==dd

True

Comparison with Table/Sum combination:

SeedRandom[1]
n = 100;
m = 50;
aa = RandomReal[1, n, m];
bb = RandomReal[1, n, m];
cc = RandomReal[1, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m]; //
 RepeatedTiming // First

0.417

ee1 = Transpose[aa].( bb cc) ; // RepeatedTiming // First

0.000051

ee2 = Transpose[aa cc].bb ; // RepeatedTiming // First

0.000045

dd == ee1 == ee2

True

share|improve this answer

edited Jun 16 at 4:00

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges

$endgroup$

add a comment
 | 

8

$begingroup$

n = 4;
m = 3;
aa = Array[a, n, m];
bb = Array[b, n, m];
cc = Array[c, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m];

You can Transpose aa and Dot it with bb cc:

Transpose[aa].(bb cc) == dd

True

Alternatively, you can Transpose aa cc and Dot it with bb:

Transpose[aa cc].bb ==dd

True

Comparison with Table/Sum combination:

SeedRandom[1]
n = 100;
m = 50;
aa = RandomReal[1, n, m];
bb = RandomReal[1, n, m];
cc = RandomReal[1, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m]; //
 RepeatedTiming // First

0.417

ee1 = Transpose[aa].( bb cc) ; // RepeatedTiming // First

0.000051

ee2 = Transpose[aa cc].bb ; // RepeatedTiming // First

0.000045

dd == ee1 == ee2

True

share|improve this answer

edited Jun 16 at 4:00

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

n = 4;
m = 3;
aa = Array[a, n, m];
bb = Array[b, n, m];
cc = Array[c, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m];

You can Transpose aa and Dot it with bb cc:

Transpose[aa].(bb cc) == dd

True

Alternatively, you can Transpose aa cc and Dot it with bb:

Transpose[aa cc].bb ==dd

True

Comparison with Table/Sum combination:

SeedRandom[1]
n = 100;
m = 50;
aa = RandomReal[1, n, m];
bb = RandomReal[1, n, m];
cc = RandomReal[1, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m]; //
 RepeatedTiming // First

0.417

ee1 = Transpose[aa].( bb cc) ; // RepeatedTiming // First

0.000051

ee2 = Transpose[aa cc].bb ; // RepeatedTiming // First

0.000045

dd == ee1 == ee2

True

share|improve this answer

edited Jun 16 at 4:00

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges

$endgroup$

n = 4;
m = 3;
aa = Array[a, n, m];
bb = Array[b, n, m];
cc = Array[c, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m];

Transpose[aa].(bb cc) == dd

Transpose[aa cc].bb ==dd

SeedRandom[1]
n = 100;
m = 50;
aa = RandomReal[1, n, m];
bb = RandomReal[1, n, m];
cc = RandomReal[1, n];
dd = Table[Sum[aa[[i, j]] bb[[i, k]] cc[[i]], i, 1, n], j, 1, m, k, 1, m]; //
 RepeatedTiming // First

ee1 = Transpose[aa].( bb cc) ; // RepeatedTiming // First

ee2 = Transpose[aa cc].bb ; // RepeatedTiming // First

dd == ee1 == ee2

share|improve this answer

edited Jun 16 at 4:00

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges

answered Jun 16 at 3:37

kglrkglr

224k1010 gold badges254254 silver badges511511 bronze badges