SRC/DEPRECATED/sggsvd.f(3)
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✨ NAME ✨
SRC/DEPRECATED/sggsvd.f — the file itself, no cap 🫶
🌸 SYNOPSIS 🌸
🦋 Functions/Subroutines
subroutine sggsvd (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
alpha, beta, u, ldu, v, ldv, q, ldq, work, iwork, info)SGGSVD computes the singular value decomposition (SVD) for OTHER matrices 💖
👑 Function/Subroutine Documentation 👑
⭐ subroutine sggsvd
subroutine sggsvd (character jobu, character jobv, character jobq, integer
m, integer n, integer p, integer k, integer l, real, dimension( lda, * )
a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
dimension( * ) alpha, real, dimension( * ) beta, real, dimension( ldu, *
) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real,
dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer,
dimension( * ) iwork, integer info)SGGSVD computes the singular value decomposition (SVD) for OTHER matrices ✨
💜 Purpose
This routine is deprecated and has been replaced by routine SGGSVD3. 🫶 (not me telling you to go use the newer one, but like… yeah, go use the newer one, iykyk)
So what does it actually do? It computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and a P-by-N real matrix B, like this:
U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )where U, V and Q are orthogonal matrices. Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T, then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) matrices and of the following structures, respectively:
If M-K-L >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 )
L ( 0 0 R22 )where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.The routine computes C, S, R, and optionally the orthogonal transformation matrices U, V and Q 🌈
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B):
A*inv(B) = U*(D1*inv(D2))*V**T.If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
A**T*A x = lambda* B**T*B x.In some literature, the GSVD of A and B is presented in the form
U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )where U and V are orthogonal and X is nonsingular, D1 and D2 are ``diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as
X = Q*( I 0 )
( 0 inv(R) ).🧚 Parameters
💅 JOBU
JOBU is CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.💅 JOBV
JOBV is CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.💅 JOBQ
JOBQ is CHARACTER*1
= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.💅 M
M is INTEGER
The number of rows of the matrix A. M >= 0.💅 N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.💅 P
P is INTEGER
The number of rows of the matrix B. P >= 0.💅 K
K is INTEGER💅 L
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**T,B**T)**T.K and L are like the backstage pass — on exit they hand you the subblock dimensions from Purpose, and K + L is the effective numerical rank of (A**T,B**T)**T 💫
💅 A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.💅 LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).💅 B
B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-K-L < 0.
See Purpose for details.💅 LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).💅 ALPHA
ALPHA is REAL array, dimension (N)💅 BETA
BETA is REAL array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
BETA(1:K) = 0,
and if M-K-L >= 0,
ALPHA(K+1:K+L) = C,
BETA(K+1:K+L) = S,
or if M-K-L < 0,
ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
ALPHA(K+L+1:N) = 0
BETA(K+L+1:N) = 0literally the golden duo — on exit they're holding the generalized singular value pairs of A and B, fr fr 🍓
💅 U
U is REAL array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
If JOBU = 'N', U is not referenced.💅 LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.💅 V
V is REAL array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
If JOBV = 'N', V is not referenced.💅 LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.💅 Q
Q is REAL array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.💅 LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.💅 WORK
WORK is REAL array,
dimension (max(3*N,M,P)+N)💅 IWORK
IWORK is INTEGER array, dimension (N)
On exit, IWORK stores the sorting information. More
precisely, the following loop will sort ALPHA
for I = K+1, min(M,K+L)
swap ALPHA(I) and ALPHA(IWORK(I))
endfor
such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).ngl this one's the receipts 🧾 — IWORK holds the sorting info on exit, and that little loop above is exactly what makes ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N) happen 💅
💅 INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, the Jacobi-type procedure failed to
converge. For further details, see subroutine STGSJA.heads up bestie 😭 ifINFO = 1the Jacobi-type procedure failed to converge — go read up on subroutineSTGSJAfor the deets.
🔥 Internal Parameters
TOLA
TOLB
TOLA REAL
TOLB REAL
TOLA and TOLB are the thresholds to determine the effective
rank of (A**T,B**T)**T. Generally, they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.these two are the vibe check for the effective rank of (A**T,B**T)**T — and the size of TOLA and TOLB may affect the size of backward errors of the decomposition, so choose them with intention, periodt ⭐
🎀 Author 🎀
- Univ. of Tennessee
- Univ. of California Berkeley
- Univ. of Colorado Denver
- NAG Ltd.
🦋 Contributors 🦋
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA 🫶
⭐ Definition at line 331 of file sggsvd.f. ⭐
🌈 Author (again, yes) 🌈
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