DCHKBD
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
DCHKBD checks the singular value decomposition (SVD) routines.
DGEBRD reduces a real general m by n matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q' * A * P = B
(or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n
and lower bidiagonal if m < n.
DORGBR generates the orthogonal matrices Q and P' from DGEBRD.
Note that Q and P are not necessarily square.
DBDSQR computes the singular value decomposition of the bidiagonal
matrix B as B = U S V'. It is called three times to compute
1) B = U S1 V', where S1 is the diagonal matrix of singular
values and the columns of the matrices U and V are the left
and right singular vectors, respectively, of B.
2) Same as 1), but the singular values are stored in S2 and the
singular vectors are not computed.
3) A = (UQ) S (P'V'), the SVD of the original matrix A.
In addition, DBDSQR has an option to apply the left orthogonal matrix
U to a matrix X, useful in least squares applications.
DBDSDC computes the singular value decomposition of the bidiagonal
matrix B as B = U S V' using divide-and-conquer. It is called twice
to compute
1) B = U S1 V', where S1 is the diagonal matrix of singular
values and the columns of the matrices U and V are the left
and right singular vectors, respectively, of B.
2) Same as 1), but the singular values are stored in S2 and the
singular vectors are not computed.
For each pair of matrix dimensions (M,N) and each selected matrix
type, an M by N matrix A and an M by NRHS matrix X are generated.
The problem dimensions are as follows
A: M x N
Q: M x min(M,N) (but M x M if NRHS > 0)
P: min(M,N) x N
B: min(M,N) x min(M,N)
U, V: min(M,N) x min(M,N)
S1, S2 diagonal, order min(M,N)
X: M x NRHS
For each generated matrix, 14 tests are performed:
Test DGEBRD and DORGBR
(1) | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
(2) | I - Q' Q | / ( M ulp )
(3) | I - PT PT' | / ( N ulp )
Test DBDSQR on bidiagonal matrix B
(4) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
(5) | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X
and Z = U' Y.
(6) | I - U' U | / ( min(M,N) ulp )
(7) | I - VT VT' | / ( min(M,N) ulp )
(8) S1 contains min(M,N) nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(9) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
computing U and V.
(10) 0 if the true singular values of B are within THRESH of
those in S1. 2*THRESH if they are not. (Tested using
DSVDCH)
Test DBDSQR on matrix A
(11) | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
(12) | X - (QU) Z | / ( |X| max(M,k) ulp )
(13) | I - (QU)'(QU) | / ( M ulp )
(14) | I - (VT PT) (PT'VT') | / ( N ulp )
Test DBDSDC on bidiagonal matrix B
(15) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
(16) | I - U' U | / ( min(M,N) ulp )
(17) | I - VT VT' | / ( min(M,N) ulp )
(18) S1 contains min(M,N) nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(19) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
computing U and V.
The possible matrix types are
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (3), but multiplied by SQRT( overflow threshold )
(7) Same as (3), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U D V, where U and V are orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U D V, where U and V are orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U D V, where U and V are orthogonal and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Rectangular matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
Special case:
(16) A bidiagonal matrix with random entries chosen from a
logarithmic distribution on [ulp^2,ulp^(-2)] (I.e., each
entry is e^x, where x is chosen uniformly on
[ 2 log(ulp), -2 log(ulp) ] .) For *this* type:
(a) DGEBRD is not called to reduce it to bidiagonal form.
(b) the bidiagonal is min(M,N) x min(M,N); if M
matrix will be lower bidiagonal, otherwise upper.
(c) only tests 5--8 and 14 are performed.
A subset of the full set of matrix types may be selected through
the logical array DOTYPE.
DGEBRD reduces a real general m by n matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q' * A * P = B
(or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n
and lower bidiagonal if m < n.
DORGBR generates the orthogonal matrices Q and P' from DGEBRD.
Note that Q and P are not necessarily square.
DBDSQR computes the singular value decomposition of the bidiagonal
matrix B as B = U S V'. It is called three times to compute
1) B = U S1 V', where S1 is the diagonal matrix of singular
values and the columns of the matrices U and V are the left
and right singular vectors, respectively, of B.
2) Same as 1), but the singular values are stored in S2 and the
singular vectors are not computed.
3) A = (UQ) S (P'V'), the SVD of the original matrix A.
In addition, DBDSQR has an option to apply the left orthogonal matrix
U to a matrix X, useful in least squares applications.
DBDSDC computes the singular value decomposition of the bidiagonal
matrix B as B = U S V' using divide-and-conquer. It is called twice
to compute
1) B = U S1 V', where S1 is the diagonal matrix of singular
values and the columns of the matrices U and V are the left
and right singular vectors, respectively, of B.
2) Same as 1), but the singular values are stored in S2 and the
singular vectors are not computed.
For each pair of matrix dimensions (M,N) and each selected matrix
type, an M by N matrix A and an M by NRHS matrix X are generated.
The problem dimensions are as follows
A: M x N
Q: M x min(M,N) (but M x M if NRHS > 0)
P: min(M,N) x N
B: min(M,N) x min(M,N)
U, V: min(M,N) x min(M,N)
S1, S2 diagonal, order min(M,N)
X: M x NRHS
For each generated matrix, 14 tests are performed:
Test DGEBRD and DORGBR
(1) | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
(2) | I - Q' Q | / ( M ulp )
(3) | I - PT PT' | / ( N ulp )
Test DBDSQR on bidiagonal matrix B
(4) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
(5) | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X
and Z = U' Y.
(6) | I - U' U | / ( min(M,N) ulp )
(7) | I - VT VT' | / ( min(M,N) ulp )
(8) S1 contains min(M,N) nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(9) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
computing U and V.
(10) 0 if the true singular values of B are within THRESH of
those in S1. 2*THRESH if they are not. (Tested using
DSVDCH)
Test DBDSQR on matrix A
(11) | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
(12) | X - (QU) Z | / ( |X| max(M,k) ulp )
(13) | I - (QU)'(QU) | / ( M ulp )
(14) | I - (VT PT) (PT'VT') | / ( N ulp )
Test DBDSDC on bidiagonal matrix B
(15) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
(16) | I - U' U | / ( min(M,N) ulp )
(17) | I - VT VT' | / ( min(M,N) ulp )
(18) S1 contains min(M,N) nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(19) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
computing U and V.
The possible matrix types are
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (3), but multiplied by SQRT( overflow threshold )
(7) Same as (3), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U D V, where U and V are orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U D V, where U and V are orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U D V, where U and V are orthogonal and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Rectangular matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
Special case:
(16) A bidiagonal matrix with random entries chosen from a
logarithmic distribution on [ulp^2,ulp^(-2)] (I.e., each
entry is e^x, where x is chosen uniformly on
[ 2 log(ulp), -2 log(ulp) ] .) For *this* type:
(a) DGEBRD is not called to reduce it to bidiagonal form.
(b) the bidiagonal is min(M,N) x min(M,N); if M
(c) only tests 5--8 and 14 are performed.
A subset of the full set of matrix types may be selected through
the logical array DOTYPE.
Arguments
| NSIZES |
(input) INTEGER
The number of values of M and N contained in the vectors
MVAL and NVAL. The matrix sizes are used in pairs (M,N). |
| MVAL |
(input) INTEGER array, dimension (NM)
The values of the matrix row dimension M.
|
| NVAL |
(input) INTEGER array, dimension (NM)
The values of the matrix column dimension N.
|
| NTYPES |
(input) INTEGER
The number of elements in DOTYPE. If it is zero, DCHKBD
does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrices are in A and B. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . |
| DOTYPE |
(input) LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix
of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. |
| NRHS |
(input) INTEGER
The number of columns in the "right-hand side" matrices X, Y,
and Z, used in testing DBDSQR. If NRHS = 0, then the operations on the right-hand side will not be tested. NRHS must be at least 0. |
| ISEED |
(input/output) INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The values of ISEED are changed on exit, and can be used in the next call to DCHKBD to continue the same random number sequence. |
| THRESH |
(input) DOUBLE PRECISION
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. Note that the expected value of the test ratios is O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. |
| A |
(workspace) DOUBLE PRECISION array, dimension (LDA,NMAX)
where NMAX is the maximum value of N in NVAL.
|
| LDA |
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,MMAX),
where MMAX is the maximum value of M in MVAL. |
| BD |
(workspace) DOUBLE PRECISION array, dimension
(max(min(MVAL(j),NVAL(j))))
|
| BE |
(workspace) DOUBLE PRECISION array, dimension
(max(min(MVAL(j),NVAL(j))))
|
| S1 |
(workspace) DOUBLE PRECISION array, dimension
(max(min(MVAL(j),NVAL(j))))
|
| S2 |
(workspace) DOUBLE PRECISION array, dimension
(max(min(MVAL(j),NVAL(j))))
|
| X |
(workspace) DOUBLE PRECISION array, dimension (LDX,NRHS)
|
| LDX |
(input) INTEGER
The leading dimension of the arrays X, Y, and Z.
LDX >= max(1,MMAX) |
| Y |
(workspace) DOUBLE PRECISION array, dimension (LDX,NRHS)
|
| Z |
(workspace) DOUBLE PRECISION array, dimension (LDX,NRHS)
|
| Q |
(workspace) DOUBLE PRECISION array, dimension (LDQ,MMAX)
|
| LDQ |
(input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,MMAX).
|
| PT |
(workspace) DOUBLE PRECISION array, dimension (LDPT,NMAX)
|
| LDPT |
(input) INTEGER
The leading dimension of the arrays PT, U, and V.
LDPT >= max(1, max(min(MVAL(j),NVAL(j)))). |
| U |
(workspace) DOUBLE PRECISION array, dimension
(LDPT,max(min(MVAL(j),NVAL(j))))
|
| V |
(workspace) DOUBLE PRECISION array, dimension
(LDPT,max(min(MVAL(j),NVAL(j))))
|
| WORK |
(workspace) DOUBLE PRECISION array, dimension (LWORK)
|
| LWORK |
(input) INTEGER
The number of entries in WORK. This must be at least
3(M+N) and M(M + max(M,N,k) + 1) + N*min(M,N) for all pairs (M,N)=(MM(j),NN(j)) |
| IWORK |
(workspace) INTEGER array, dimension at least 8*min(M,N)
|
| NOUT |
(input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.) |
| INFO |
(output) INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0 -2: Some MM(j) < 0 -3: Some NN(j) < 0 -4: NTYPES < 0 -6: NRHS < 0 -8: THRESH < 0 -11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ). -17: LDB < 1 or LDB < MMAX. -21: LDQ < 1 or LDQ < MMAX. -23: LDPT< 1 or LDPT< MNMAX. -27: LWORK too small. If DLATMR, SLATMS, DGEBRD, DORGBR, or DBDSQR, returns an error code, the absolute value of it is returned. *----------------------------------------------------------------------- Some Local Variables and Parameters: ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. MMAX Largest value in NN. NMAX Largest value in NN. MNMIN min(MM(j), NN(j)) (the dimension of the bidiagonal matrix.) MNMAX The maximum value of MNMIN for j=1,...,NSIZES. NFAIL The number of tests which have exceeded THRESH COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. RTOVFL, RTUNFL Square roots of the previous 2 values. ULP, ULPINV Finest relative precision and its inverse. The following four arrays decode JTYPE: KTYPE(j) The general type (1-10) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) |