svdappend
Description
Examples
Revise SVD with New Columns
Calculate the SVD of a 5-by-5 random matrix, and then calculate the revised SVD with three additional columns.
Create a 5-by-5 random matrix and calculate the SVD.
A = rand(5); [U,S,V] = svd(A)
U = 5×5
-0.2475 -0.5600 0.4131 0.5759 0.3504
-0.3542 -0.5207 -0.7577 -0.0111 -0.1707
-0.4641 0.6013 -0.1679 0.6063 -0.1652
-0.5475 -0.1183 0.4755 -0.3314 -0.5919
-0.5460 0.1992 -0.0298 -0.4369 0.6859
S = 5×5
3.3129 0 0 0 0
0 0.9431 0 0 0
0 0 0.8358 0 0
0 0 0 0.4837 0
0 0 0 0 0.0198
V = 5×5
-0.4307 -0.8839 0.0530 -0.0884 0.1503
-0.4309 0.2207 0.1961 -0.7322 -0.4370
-0.4617 0.0890 -0.7467 0.3098 -0.3539
-0.4730 0.3701 -0.0798 -0.1023 0.7890
-0.4380 0.1585 0.6283 0.5913 -0.1968
Create a 5-by-3 matrix containing three new columns of data, and then calculate the revised SVD.
D = rand(5,3); [U1,S1,V1] = svdappend(U,S,V,D)
U1 = 5×5
0.3616 0.6757 -0.4056 0.3016 0.3966
0.4066 0.2922 0.8635 0.0546 -0.0270
0.4272 -0.3990 -0.1227 0.7092 -0.3745
0.5707 0.0815 -0.2734 -0.6000 -0.4826
0.4424 -0.5406 0.0091 -0.2075 0.6848
S1 = 5×1
3.7334
1.3862
0.8863
0.6852
0.4189
V1 = 8×5
0.4066 0.3586 0.2168 -0.4290 0.5809
0.3631 -0.3711 -0.1345 -0.4995 0.0597
0.3995 -0.2777 0.5995 0.4701 -0.0200
0.3993 -0.4331 -0.0153 0.0597 -0.0557
0.3878 -0.1270 -0.6640 0.1470 -0.1071
0.3197 0.3849 0.1225 0.1730 -0.1565
0.1221 0.2360 -0.3436 0.5301 0.5243
0.3410 0.4990 -0.0366 -0.0964 -0.5870
Control Size of Revised SVD
Calculate the SVD of a 6-by-6 random matrix, and then calculate the revised SVD with four additional rows. Use the MaxRank
and WindowSize
name-value arguments to control the calculation and size of the output.
Create a 6-by-6 random matrix and calculate the SVD.
A = rand(6); [U,S,V] = svd(A)
U = 6×6
-0.4758 -0.0015 -0.4257 0.6098 -0.4496 -0.1358
-0.4440 -0.4208 -0.1279 -0.2969 0.3761 -0.6163
-0.4229 0.2502 0.2709 -0.5781 -0.5923 -0.0053
-0.2832 -0.4610 0.7395 0.3449 0.0073 0.2035
-0.3375 -0.2843 -0.3978 -0.2476 0.1740 0.7453
-0.4510 0.6834 0.1544 0.1598 0.5247 0.0695
S = 6×6
3.5693 0 0 0 0 0
0 1.2230 0 0 0 0
0 0 1.0101 0 0 0
0 0 0 0.6166 0 0
0 0 0 0 0.4062 0
0 0 0 0 0 0.0390
V = 6×6
-0.3809 -0.7234 0.0105 0.5327 0.1649 -0.1430
-0.4327 0.1494 0.8629 -0.1576 0.1404 0.0352
-0.4497 0.3557 -0.1725 0.1099 -0.4108 -0.6787
-0.5038 0.1141 -0.4450 -0.3723 0.6261 0.0674
-0.3875 -0.3141 -0.1275 -0.3898 -0.6244 0.4394
-0.2478 0.4650 -0.1059 0.6235 -0.0496 0.5657
Create a 4-by-6 matrix containing four new rows of data, and then calculate the revised SVD. Specify Shape="rows"
to append the data in D
as rows.
D = rand(4,6);
[U1,S1,V1] = svdappend(U,S,V,D,Shape="rows")
U1 = 10×6
0.3909 0.0593 -0.3501 -0.2543 0.2971 0.6734
0.3588 0.2925 -0.2515 0.3982 0.1097 -0.3212
0.3429 -0.3027 0.1746 0.3867 -0.2745 0.3894
0.2353 0.4938 0.5672 0.2279 -0.0616 0.0268
0.2715 0.1703 -0.4510 0.2314 0.0597 -0.1402
0.3767 -0.5450 0.2275 -0.2637 0.1615 -0.2770
0.3299 0.1748 -0.1556 -0.4787 -0.7600 -0.1485
0.2389 -0.1394 -0.1109 -0.0742 0.2814 -0.4153
0.2908 0.3004 0.3953 -0.3542 0.3391 0.0045
0.2818 -0.3305 0.1186 0.3035 -0.1384 0.0052
S1 = 6×1
4.3460
1.3335
1.1287
1.0049
0.5681
0.4734
V1 = 6×6
0.3915 0.8486 0.0006 -0.1502 0.3210 -0.0307
0.4490 -0.1113 0.8111 0.2481 -0.1583 -0.2038
0.4156 -0.3870 -0.0439 0.0599 0.5980 0.5608
0.4960 -0.2554 -0.4951 0.1048 0.0576 -0.6552
0.3582 0.1589 -0.3070 0.3938 -0.6290 0.4489
0.3132 -0.1649 0.0298 -0.8638 -0.3396 0.1108
Specify MaxRank=5
to impose a limit on the number of singular values in S2
and the number of singular vectors in U2
and V2
. After calculating the revised SVD, svdappend
truncates the smallest singular values and associated singular vectors until the limit is satisfied. These results are similar to U1
, S1
, and V1
, but the smallest singular value and associated singular vectors are truncated.
[U2,S2,V2] = svdappend(U,S,V,D,Shape="rows",MaxRank=5)
U2 = 10×5
0.3909 0.0593 -0.3501 -0.2543 0.2971
0.3588 0.2925 -0.2515 0.3982 0.1097
0.3429 -0.3027 0.1746 0.3867 -0.2745
0.2353 0.4938 0.5672 0.2279 -0.0616
0.2715 0.1703 -0.4510 0.2314 0.0597
0.3767 -0.5450 0.2275 -0.2637 0.1615
0.3299 0.1748 -0.1556 -0.4787 -0.7600
0.2389 -0.1394 -0.1109 -0.0742 0.2814
0.2908 0.3004 0.3953 -0.3542 0.3391
0.2818 -0.3305 0.1186 0.3035 -0.1384
S2 = 5×1
4.3460
1.3335
1.1287
1.0049
0.5681
V2 = 6×5
0.3915 0.8486 0.0006 -0.1502 0.3210
0.4490 -0.1113 0.8111 0.2481 -0.1583
0.4156 -0.3870 -0.0439 0.0599 0.5980
0.4960 -0.2554 -0.4951 0.1048 0.0576
0.3582 0.1589 -0.3070 0.3938 -0.6290
0.3132 -0.1649 0.0298 -0.8638 -0.3396
Specify WindowSize=5
to instead discard some of the old rows of data before calculating the revised SVD. Because you specified four new rows of data, this calculation uses those rows and one old row to calculate the revised SVD. Therefore, the results are different than U2
, S2
, and V2
, even though both calculations return five singular values.
[U3,S3,V3] = svdappend(U,S,V,D,Shape="rows",WindowSize=5)
U3 = 5×5
0.5744 0.4555 -0.1212 -0.5750 -0.3424
0.4696 -0.5545 0.6641 -0.0052 -0.1761
0.3423 0.1068 0.0897 -0.1393 0.9187
0.4155 -0.5223 -0.7285 0.1546 0.0005
0.3997 0.4482 0.0745 0.7912 -0.0883
S3 = 5×1
3.0549
1.0150
0.6496
0.4210
0.3188
V3 = 6×5
0.2944 -0.7761 -0.3257 0.1669 0.4170
0.5009 0.1933 -0.4472 0.3182 -0.4043
0.4010 0.4166 -0.3378 -0.0369 0.1463
0.4957 0.2692 0.4236 -0.1555 0.5918
0.2325 -0.1051 0.5839 0.6846 -0.2150
0.4494 -0.3212 0.2439 -0.6138 -0.4948
Specify Error Tolerances for Singular Values
Calculate the SVD of a 6-by-6 test matrix, and then calculate the revised SVD with two additional columns. Use the AbsoluteTolerance
and RelativeTolerance
name-value arguments to filter out small singular values and their associated singular vectors.
Create a 6-by-6 test matrix that has only one large singular value using the gallery
function. Calculate the SVD.
rng default A = gallery("randsvd",6,[],1); [U,S,V] = svd(A,"vector")
U = 6×6
-0.5231 0.4398 -0.2661 -0.0608 -0.0294 0.6765
-0.1183 -0.2117 0.7924 -0.1320 0.4046 0.3636
0.2745 -0.2283 -0.1911 -0.8821 -0.1331 0.2005
-0.4983 -0.5383 -0.4065 -0.0453 0.5131 -0.1770
-0.6224 0.0395 0.3152 -0.2965 -0.4881 -0.4308
0.0372 0.6468 0.0153 -0.3329 0.5623 -0.3912
S = 6×1
1.0000
0.0000
0.0000
0.0000
0.0000
0.0000
V = 6×6
-0.2745 0.5572 -0.1389 -0.6956 -0.1745 0.2839
0.0787 0.4188 0.3088 0.5010 -0.6452 0.2363
-0.0477 -0.2451 0.6644 -0.4136 -0.3366 -0.4602
0.5721 0.5285 -0.1578 -0.0377 0.0885 -0.5994
-0.5411 -0.0578 -0.5104 0.1651 -0.4039 -0.5030
0.5442 -0.4141 -0.3981 -0.2557 -0.5187 0.1987
Create a 6-by-2 matrix containing two new columns of data, and then calculate the revised SVD. The vector of singular values S1
now contains three large singular values.
D = rand(6,2); [U1,S1,V1] = svdappend(U,S,V,D)
U1 = 6×6
0.5419 -0.0980 -0.2440 -0.2845 -0.3320 -0.6679
0.4939 0.4285 0.2632 0.6155 0.3072 -0.1732
0.0935 0.4736 0.6259 -0.4188 -0.4163 0.1630
0.4448 -0.2781 0.1769 -0.4897 0.6326 0.2311
0.4456 -0.4660 0.0556 0.3207 -0.4706 0.5069
0.2395 0.5366 -0.6672 -0.1491 -0.0193 0.4324
S1 = 6×1
2.1458
0.6808
0.3284
0.0000
0.0000
0.0000
V1 = 8×6
0.1031 -0.2335 -0.1009 -0.2977 -0.7510 -0.0757
-0.0296 0.0670 0.0289 0.5449 -0.4660 -0.4599
0.0179 -0.0406 -0.0175 0.5849 -0.2703 0.6476
-0.2149 0.4866 0.2103 -0.1370 -0.1525 0.4882
0.2033 -0.4603 -0.1989 -0.3399 -0.1447 0.3425
-0.2045 0.4630 0.2001 -0.3716 -0.3186 -0.0876
0.6506 0.5247 -0.5490 -0.0000 0.0000 0.0000
0.6599 -0.0330 0.7506 0.0000 -0.0000 -0.0000
Now, perform the same calculation but specify AbsoluteTolerance=1e-6
to filter out any singular values smaller than that threshold.
[U2,S2,V2] = svdappend(U,S,V,D,AbsoluteTolerance=1e-6)
U2 = 6×3
0.5419 -0.0980 -0.2440
0.4939 0.4285 0.2632
0.0935 0.4736 0.6259
0.4448 -0.2781 0.1769
0.4456 -0.4660 0.0556
0.2395 0.5366 -0.6672
S2 = 3×1
2.1458
0.6808
0.3284
V2 = 8×3
0.1031 -0.2335 -0.1009
-0.0296 0.0670 0.0289
0.0179 -0.0406 -0.0175
-0.2149 0.4866 0.2103
0.2033 -0.4603 -0.1989
-0.2045 0.4630 0.2001
0.6506 0.5247 -0.5490
0.6599 -0.0330 0.7506
Alternatively, specify RelativeTolerance=0.3
to filter out singular values smaller than 30% of the largest singular value. While the absolute tolerance can be any number, the relative tolerance is a value from 0 to 1 indicating how small a singular value can be relative to the largest singular value.
[U3,S3,V3] = svdappend(U,S,V,D,RelativeTolerance=0.3)
U3 = 6×2
0.5419 -0.0980
0.4939 0.4285
0.0935 0.4736
0.4448 -0.2781
0.4456 -0.4660
0.2395 0.5366
S3 = 2×1
2.1458
0.6808
V3 = 8×2
0.1031 -0.2335
-0.0296 0.0670
0.0179 -0.0406
-0.2149 0.4866
0.2033 -0.4603
-0.2045 0.4630
0.6506 0.5247
0.6599 -0.0330
Input Arguments
U,S,V
— Existing SVD factors
matrices
Existing SVD factors, specified as matrices. U
,
S
, and V
can be SVD factors of any shape
calculated by one of these functions:
How you should recombine the SVD factors differs depending on whether the singular
values are returned as a vector or matrix, and whether the decomposition is complete,
economy size, or truncated. For example, if S
is a vector returned as
part of a complete decomposition of a square matrix A
, then
A = (U.*S')*V'
.
Example: [U,S,V] = svd(A)
Example: [U,S,V] = svd(A,"vector")
Example: [U,S,V] = svd(A,"econ")
Example: [U,S,V] = svd(A,"econ","vector")
Example: [U,S,V] = svds(A,5)
Example: [U,S,V] = svdsketch(A,1e-5)
Data Types: single
| double
Complex Number Support: Yes
D
— New columns or rows to append
vector | matrix
New columns or rows to append, specified as a vector or matrix. Use the
Shape
name-value argument to specify whether D
contains columns or rows of data to append. By default, svdappend
appends D
as new columns of data.
When U
, S
, and V
are
nonempty and D
has a different data type than those factors,
svdappend
casts D
to be the same data
type.
Data Types: single
| double
Complex Number Support: Yes
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Example: [U1,S1,V1] =
svdappend(U,S,V,D,Shape="rows",WindowSize=10)
Shape
— Shape of data in D
"columns"
(default) | "rows"
Shape of data in D
, specified as "columns"
or "rows"
. If the value is "columns"
, the
revised SVD is for [A D]
. If the value is
"rows"
, the revised SVD is for [A; D]
. Neither
[A D]
nor [A; D]
are explicitly formed.
Example: [U1,S1,V1] =
svdappend(U,S,V,D,Shape="rows")
WindowSize
— Maximum number of columns or rows to compute SVD
Inf
(default) | positive integer scalar
Maximum number of columns or rows to compute the SVD, specified as a positive
integer scalar. WindowSize
controls the memory consumption of the
algorithm. The function appends new data and discards the oldest data (starting with
the first column or row) to satisfy WindowSize
before computing the
revised SVD. By default, svdappend
does not discard data.
Example: [U1,S1,V1] =
svdappend(U,S,V,D,WindowSize=50)
AbsoluteTolerance
— Absolute tolerance for singular values and vectors
nonnegative real scalar
Absolute tolerance for singular values and vectors, specified as a nonnegative
real scalar. The function truncates singular values in S1
less than
or equal to AbsoluteTolerance
. Corresponding singular vectors in
U1
and V1
are also truncated. By default,
svdappend
does not truncate the outputs.
Each of the name-value arguments AbsoluteTolerance
,
RelativeTolerance
, and MaxRank
result in a
specific rank for the revised SVD. svdappend
truncates the
revised SVD based on the minimum rank from all specified arguments.
Example: [U1,S1,V1] =
svdappend(U,S,V,D,AbsoluteTolerance=1e-6)
Data Types: single
| double
RelativeTolerance
— Relative tolerance for singular values and vectors
real scalar in range [0 1]
Relative tolerance for singular values and vectors, specified as a real scalar in
the range [0 1]
. The function truncates singular values in
S1
less than or equal to
RelativeTolerance*S1(1)
. Corresponding singular vectors in
U1
and V1
are also truncated. By default,
svdappend
does not truncate the outputs.
Each of the name-value arguments AbsoluteTolerance
,
RelativeTolerance
, and MaxRank
result in a
specific rank for the revised SVD. svdappend
truncates the
revised SVD based on the minimum rank from all specified arguments.
Example: [U1,S1,V1] =
svdappend(U,S,V,D,RelativeTolerance=0.3)
Data Types: single
| double
MaxRank
— Maximum rank of revised SVD
positive integer scalar
Maximum rank of the revised SVD, specified as a positive integer scalar.
MaxRank
imposes a limit on the number of singular values in
S1
and the number of singular vectors in U1
and V1
. By default, there is no limit to the rank.
Each of the name-value arguments AbsoluteTolerance
,
RelativeTolerance
, and MaxRank
result in a
specific rank for the revised SVD. svdappend
truncates the
revised SVD based on the minimum rank from all specified arguments.
Example: [U1,S1,V1] =
svdappend(U,S,V,D,MaxRank=25)
Data Types: single
| double
Output Arguments
U1
, V1
— Revised singular vectors
matrices
Revised singular vectors, returned as matrices.
If
Shape="columns"
, then(U1.*S1')*V1'
is equivalent to[U*S*V' D]
up to round-off error, ignoring truncation.If
Shape="rows"
, then(U1.*S1')*V1'
is equivalent to[U*S*V'; D]
up to round-off error, ignoring truncation.Each of the name-value arguments
AbsoluteTolerance
,RelativeTolerance
, andMaxRank
result in a specific rank for the revised SVD. Use these arguments to truncate singular values and vectors to control the rank ofU1*diag(S1)*V1'
.
Different machines and releases of MATLAB® can produce different singular vectors that are still numerically
accurate. Corresponding columns in U1
and V1
can
flip their signs, since this change does not affect the value of the expression
U1*S1*V1'
.
S1
— Revised singular values
column vector
Revised singular values, returned as a column vector. Use
diag(S1)
to create a diagonal matrix of singular values from
S1
.
Version History
Introduced in R2023b
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