DCHKBD(3)

📦 version 3.12.0, in opensuse-leap 16.0

dchkbd.f(3) 💅 Library Functions Manual — ok bestie, this is the dchkbd.f(3) page for the LAPACK test routine at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/dchkbd.f. Yes, the whole path is the NAME, no cap 💖

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/dchkbd.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine DCHKBD (nsizes, mval, nval, ntypes, dotype, nrhs, iseed,
    thresh, a, lda, bd, be, s1, s2, x, ldx, y, z, q, ldq, pt, ldpt, u,
    vt, work, lwork, iwork, nout, info)
    DCHKBD

🦋 Function/Subroutine Documentation 🦋

   subroutine DCHKBD (integer nsizes, integer, dimension( * ) mval, integer,
       dimension( * ) nval, integer ntypes, logical, dimension( * ) dotype,
       integer nrhs, integer, dimension( 4 ) iseed, double precision thresh,
       double precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( * ) bd, double precision, dimension( * ) be, double
       precision, dimension( * ) s1, double precision, dimension( * ) s2,
       double precision, dimension( ldx, * ) x, integer ldx, double precision,
       dimension( ldx, * ) y, double precision, dimension( ldx, * ) z, double
       precision, dimension( ldq, * ) q, integer ldq, double precision,
       dimension( ldpt, * ) pt, integer ldpt, double precision, dimension(
       ldpt, * ) u, double precision, dimension( ldpt, * ) vt, double
       precision, dimension( * ) work, integer lwork, integer, dimension( * )
       iwork, integer nout, integer info)
       DCHKBD

🎀 Purpose 🎀

Ok so DCHKBD checks the singular value decomposition (SVD) routines. That's the whole vibe check for these routines, bestie. 💅

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.

DBDSVDX computes the singular value decomposition of the bidiagonal matrix B as B = U S V' using bisection and inverse iteration. It is called six times to compute:

  1. B = U S1 V', RANGE='A', 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. B = U S1 V', RANGE='I', 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.
  4. Same as 3), but the singular values are stored in S2 and the singular vectors are not computed.
  5. B = U S1 V', RANGE='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.
  6. Same as 5), 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.

🔥 Test DBDSVDX on bidiagonal matrix B 🔥

  • (20) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
  • (21) | I - U' U | / ( min(M,N) ulp )
  • (22) | I - VT VT' | / ( min(M,N) ulp )
  • (23) S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.)
  • (24) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without computing U and V.
  • (25) | S1 - U' B VT' | / ( |S| n ulp ) DBDSVDX('V', 'I')
  • (26) | I - U' U | / ( min(M,N) ulp )
  • (27) | I - VT VT' | / ( min(M,N) ulp )
  • (28) S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.)
  • (29) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without computing U and V.
  • (30) | S1 - U' B VT' | / ( |S1| n ulp ) DBDSVDX('V', 'V')
  • (31) | I - U' U | / ( min(M,N) ulp )
  • (32) | I - VT VT' | / ( min(M,N) ulp )
  • (33) S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.)
  • (34) | 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 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 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 )
  • (16) Special case: 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<N, the 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.

🍓 Parameters 🍓

NSIZES

NSIZES is 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

  • MVAL — MVAL is an INTEGER array, dimension (NM). It holds the values of the matrix row dimension M. 💖
  • NVAL — NVAL is an INTEGER array, dimension (NM). It holds the values of the matrix column dimension N.
  • NTYPES — NTYPES is INTEGER. This is 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 — that one means use whatever matrices are already in A and B. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE.. 👑
  • DOTYPE — DOTYPE is a 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 — NRHS is INTEGER. This is the number of columns in the 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 — ISEED is an 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 — THRESH is DOUBLE PRECISION. This is 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 — A is a DOUBLE PRECISION array, dimension (LDA,NMAX) where NMAX is the maximum value of N in NVAL.
  • LDA — LDA is INTEGER. This is the leading dimension of the array A. LDA >= max(1,MMAX), where MMAX is the maximum value of M in MVAL.
  • BD — BD is a DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))).
  • BE — BE is a DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))).
  • S1 — S1 is a DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))).
  • S2 — S2 is a DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))).
  • X — X is a DOUBLE PRECISION array, dimension (LDX,NRHS).
  • LDX — LDX is INTEGER. This is the leading dimension of the arrays X, Y, and Z. LDX >= max(1,MMAX).
  • Y — Y is a DOUBLE PRECISION array, dimension (LDX,NRHS).
  • Z — Z is a DOUBLE PRECISION array, dimension (LDX,NRHS).
  • Q — Q is a DOUBLE PRECISION array, dimension (LDQ,MMAX).
  • LDQ — LDQ is INTEGER. This is the leading dimension of the array Q. LDQ >= max(1,MMAX).
  • PT — PT is a DOUBLE PRECISION array, dimension (LDPT,NMAX).
  • LDPT — LDPT is INTEGER. This is the leading dimension of the arrays PT, U, and V. LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
  • U — U is a DOUBLE PRECISION array, dimension (LDPT,max(min(MVAL(j),NVAL(j)))).
  • VT — VT is a DOUBLE PRECISION array, dimension (LDPT,max(min(MVAL(j),NVAL(j)))).
  • WORK — WORK is a DOUBLE PRECISION array, dimension (LWORK).
  • LWORK — LWORK is INTEGER. This is the number of entries in WORK. It 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 — IWORK is an INTEGER array, dimension at least 8*min(M,N).
  • NOUT — NOUT is INTEGER. This is the FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0).
  • INFO — INFO is 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 .
  • KMODE(j) — The MODE value to be passed to the matrix generator for type .
  • KMAGN(j) — The order of magnitude (O(1), O(overflow^(1/2) ), O(underflow^(1/2) ))

👑 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 489 of file dchkbd.f.

👑 Author

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