gsvd
Generalized singular value decomposition
Syntax
Description
returns the vector of generalized singular values,
sigma
= gsvd(A
,B
)sqrt(diag(C'*C)./diag(S'*S))
. When B
is square and
nonsingular, the generalized singular values, gsvd(A,B)
, correspond to
the ordinary singular values, svd(A/B)
, but they are sorted in the
opposite order. Their reciprocals are gsvd(B,A)
.
Examples
Generalized Singular Value Decomposition of Matrices
Perform a generalized singular value decomposition on two matrices and also calculate the generalized singular values.
Create a 5-by-3 matrix A
and a 3-by-3 matrix B
.
A = reshape(1:15,5,3)
A = 5×3
1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
B = magic(3)
B = 3×3
8 1 6
3 5 7
4 9 2
Perform a generalized singular value decomposition on A
and B
. The outputs include orthogonal U
and V
, a nonsingular X
, and diagonal matrices C
and S
. Because A
is rank deficient, the first diagonal element of C
is zero.
[U,V,X,C,S] = gsvd(A,B)
U = 5×5
-0.1882 0.6457 -0.4279 -0.4268 -0.4271
0.6897 0.3296 -0.4375 0.2067 0.4261
-0.6849 0.0135 -0.4470 0.2542 0.5160
0.0534 -0.3026 -0.4566 0.5787 -0.6019
0.1300 -0.6187 -0.4661 -0.6128 0.0869
V = 3×3
-0.7071 -0.6946 0.1325
0.0000 -0.1874 -0.9823
0.7071 -0.6946 0.1325
X = 3×3
-2.8284 -9.3761 -6.9346
5.6569 -8.3071 -18.3301
-2.8284 -7.2381 -29.7256
C = 5×3
0.0000 0 0
0 0.3155 0
0 0 0.9807
0 0 0
0 0 0
S = 3×3
1.0000 0 0
0 0.9489 0
0 0 0.1957
Now, perform an economy-size decomposition of A
and B
. The matrices U
and C
have different sizes, but the other output matrices are the same.
[U,V,X,C,S] = gsvd(A,B,"econ")
U = 5×3
-0.3736 -0.6457 0.4279
-0.0076 -0.3296 0.4375
0.8617 -0.0135 0.4470
-0.2063 0.3026 0.4566
-0.2743 0.6187 0.4661
V = 3×3
-0.7071 0.6946 -0.1325
0.0000 0.1874 0.9823
0.7071 0.6946 -0.1325
X = 3×3
-2.8284 9.3761 6.9346
5.6569 8.3071 18.3301
-2.8284 7.2381 29.7256
C = 3×3
0 0 0
0 0.3155 0
0 0 0.9807
S = 3×3
1.0000 0 0
0 0.9489 0
0 0 0.1957
Calculate the generalized singular values, which in this case are equal to the ratios diag(C)./diag(S)
. These values are a reordering of the ordinary singular values returned by svd(A/B)
.
sigma = gsvd(A,B)
sigma = 3×1
0.0000
0.3325
5.0123
svals = svd(A/B)
svals = 3×1
5.0123
0.3325
0.0000
Matrices with Infinite Singular Values
Examine why generalized singular values can sometimes be Inf
.
Create a 3-by-5 matrix A
and a 5-by-5 matrix B
.
A = reshape(1:15,3,5)
A = 3×5
1 4 7 10 13
2 5 8 11 14
3 6 9 12 15
B = magic(5)
B = 5×5
17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9
Perform a generalized singular value decomposition on A
and B
. The outputs include orthogonal U
and V
, a nonsingular X
, and diagonal matrices C
and S
. In this situation, the nonzero diagonal of C
is diag(C,2)
.
[U,V,X,C,S] = gsvd(A,B)
U = 3×3
0.4082 0.7178 -0.5639
-0.8165 0.0109 -0.5772
0.4082 -0.6961 -0.5906
V = 5×5
-0.4998 -0.6586 0.3195 -0.4552 -0.0843
0.1754 0.5301 0.6218 -0.5408 0.0956
-0.0423 0.1729 -0.0078 -0.0271 -0.9836
-0.3869 0.3344 -0.6610 -0.5408 0.0956
0.7536 -0.3788 -0.2725 -0.4552 -0.0843
X = 5×5
-0.2115 0.8645 10.0949 -30.7287 -4.6958
-2.4503 -14.9242 -2.1067 -28.5003 -11.6858
11.6680 2.1838 -14.8016 -26.2720 -18.6758
-9.0062 11.8759 -3.3206 -24.0436 -25.6657
0 0 10.1340 -21.8152 -32.6557
C = 3×5
0 0 0.0000 0 0
0 0 0 0.0439 0
0 0 0 0 0.7432
S = 5×5
1.0000 0 0 0 0
0 1.0000 0 0 0
0 0 1.0000 0 0
0 0 0 0.9990 0
0 0 0 0 0.6690
Find the generalized singular values, which include zeros.
sigma = gsvd(A,B)
sigma = 5×1
0
0
0.0000
0.0439
1.1109
Reversing the roles of A
and B
inverts these values, producing Inf
values.
sigma2 = gsvd(B,A)
sigma2 = 5×1
1016 ×
0.0000
0.0000
3.8953
Inf
Inf
Input Arguments
A
, B
— Input matrices (as separate arguments)
matrices
Input matrices. A
and B
must have the same
number of columns but can have different numbers of rows.
Data Types: single
| double
Complex Number Support: Yes
Output Arguments
U
, V
— Unitary matrix factors
matrices
Unitary matrix factors, returned as matrices. U
and
V
are matrices with orthogonal columns that provide a basis for the
columns of A
and B
, respectively.
The sizes of U
and V
depend on whether you
specify the "econ"
option. If A
is
m
-by-p
and B
is
n
-by-p
, then:
By default,
U
ism
-by-m
andV
isn
-by-n
.If you specify the
"econ"
option, thenU
ism
-by-min(m,p)
andV
isn
-by-min(n,p)
.
X
— Matrix factor
matrix
Matrix factor, returned as a matrix. If A
is
m
-by-p
and B
is
n
-by-p
, then X
is
p
-by-q
, where q
is the
numerical rank of [A; B]
.
A property of X
is that norm([A; B]) ==
norm(X)
because U
, V
,
C
, and S
represent only an orthogonal basis for
the matrix [A; B]
.
C
, S
— Diagonal matrix factors
matrices
Diagonal matrix factors, returned as matrices. The nonzero elements of
S
are always on its main diagonal. The nonzero elements of
C
are on the diagonal
diag(C,max(0,size(C,2)-size(C,1)))
. To retrieve the nonzero values,
use the commands sv = max(S,[],1)
and cv =
max(C,[],1)
. The generalized singular values of A
and
B
are equal to the ratio cv./sv
.
The sizes of C
and S
depend on whether you
specify the "econ"
option. If A
is
m
-by-p
and B
is
n
-by-p
, then:
By default,
C
ism
-by-q
andS
isn
-by-q
.If you specify the
"econ"
option, thenC
ismin(m,p)
-by-q
andV
ismin(n,p)
-by-q
.
In both cases, q
is the numerical rank of [A;
B]
.
sigma
— Generalized singular values
column vector
Generalized singular values, returned as a column vector. If sv =
max(S,[],1)
and cv = max(C,[],1)
, then the generalized
singular values of A
and B
are equal to the ratio
cv./sv
.
sigma
has length equal to the numerical rank of [A;
B]
and is in nondecreasing order.
Algorithms
The generalized singular value decomposition performed by the gsvd
function uses a C-S decomposition, as well as the built-in svd
and
qr
functions.
Extended Capabilities
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Version History
Introduced before R2006aR2022b: Output size based on numerical rank
With the function call [U,V,X,C,S] = gsvd(A,B)
, where
A
is m
-by-p
and
B
is n
-by-p
, the
gsvd
function returns:
X
asp
-by-q
C
asm
-by-q
S
asn
-by-q
Additionally, with one output argument the function call sigma =
gsvd(A,B)
returns a vector of the generalized singular values with length
q
.
The behavior change is that in all of these cases q
is now equal to
the numerical rank of [A; B]
. The numerical rank is calculated from the
QR factorization of [A; B]
. This change ensures that nonzero elements of
C
and S
are uniquely determined.
Previously, the value of q
was min(m+n,p)
, and
gsvd
returned extra columns (or elements) in the outputs when
[A; B]
was not full rank.
R2022b: "0" syntax not recommended for economy-size decompositions
The syntax [U,V,X,C,S] = gsvd(A,B,0)
will continue to be supported,
but is no longer recommended. Use the "econ"
option to perform
economy-size decompositions instead.
R2022a: Option for economy-size decompositions
Use the "econ"
option to calculate economy-size decompositions with
gsvd
. The functionality is the same as
gsvd(A,B,0)
.
Comando MATLAB
Hai fatto clic su un collegamento che corrisponde a questo comando MATLAB:
Esegui il comando inserendolo nella finestra di comando MATLAB. I browser web non supportano i comandi MATLAB.
Select a Web Site
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
You can also select a web site from the following list:
How to Get Best Site Performance
Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.
Americas
- América Latina (Español)
- Canada (English)
- United States (English)
Europe
- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)
- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom (English)