TESTING/EIG/cchkbd.f(3)

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version 3.12.1 · 2 versions
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✨ NAME ✨

TESTING/EIG/cchkbd.f

okay bestie, quick vibe check: you're in the Library Functions Manual page for TESTING/EIG/cchkbd.f(3), and the CCHKBD subroutine is the main character here. 💅

🎀 SYNOPSIS 🎀

🧸 Functions/Subroutines 🧸

  • subroutine cchkbd (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, rwork, nout, info) — CCHKBD
subroutine cchkbd (integer nsizes, integer, dimension( * ) mval, integer,
    dimension( * ) nval, integer ntypes, logical, dimension( * ) dotype,
    integer nrhs, integer, dimension( 4 ) iseed, real thresh, complex,
    dimension( lda, * ) a, integer lda, real, dimension( * ) bd, real,
    dimension( * ) be, real, dimension( * ) s1, real, dimension( * ) s2,
    complex, dimension( ldx, * ) x, integer ldx, complex, dimension( ldx, *
    ) y, complex, dimension( ldx, * ) z, complex, dimension( ldq, * ) q,
    integer ldq, complex, dimension( ldpt, * ) pt, integer ldpt, complex,
    dimension( ldpt, * ) u, complex, dimension( ldpt, * ) vt, complex,
    dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer
    nout, integer info)

🦋 Function/Subroutine Documentation 🦋

🌟 subroutine cchkbd 🌟

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

CCHKBD — yes, that's the icon. 👑

💖 Purpose 💖

okay so CCHKBD checks the singular value decomposition (SVD) routines. It's giving quality-control bestie energy. ✨

CGEBRD reduces a complex general m by n matrix A to real upper or lower bidiagonal form 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.

CUNGBR generates the orthogonal matrices Q and P' from CGEBRD. Note that Q and P are not necessarily square.

CBDSQR 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, CBDSQR has an option to apply the left orthogonal matrix U to a matrix X, useful in least squares applications.

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 CGEBRD and CUNGBR

  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 CBDSQR on bidiagonal matrix B

  1. | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
  2. | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X and Z = U' Y.
  3. | I - U' U | / ( min(M,N) ulp )
  4. | I - VT VT' | / ( min(M,N) ulp )
  5. S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.)
  6. 0 if the true singular values of B are within THRESH of those in S1. 2*THRESH if they are not. (Tested using SSVDCH)
  7. | S1 - S2 | / ( |S1| ulp ), where S2 is computed without computing U and V.

Test CBDSQR on matrix A

  1. | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
  2. | X - (QU) Z | / ( |X| max(M,k) ulp )
  3. | I - (QU)'(QU) | / ( M ulp )
  4. | I - (VT PT) (PT'VT') | / ( N ulp )

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 )

Special case:

  1. 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) CGEBRD 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 is INTEGER array, dimension (NM). The values of the matrix row dimension M.

NVAL

NVAL is INTEGER array, dimension (NM). The values of the matrix column dimension N.

NTYPES

NTYPES is INTEGER. The number of elements in DOTYPE. If it is zero, CCHKBD 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

DOTYPE is 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. The number of columns in the matrices X, Y, and Z, used in testing CBDSQR. If NRHS = 0, then the operations on the right-hand side will not be tested. NRHS must be at least 0.

ISEED

ISEED is 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 CCHKBD to continue the same random number sequence.

THRESH

THRESH is REAL. 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 COMPLEX array, dimension (LDA,NMAX) where NMAX is the maximum value of N in NVAL.

LDA

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,MMAX), where MMAX is the maximum value of M in MVAL.

BD

BD is REAL array, dimension (max(min(MVAL(j),NVAL(j))))

BE

BE is REAL array, dimension (max(min(MVAL(j),NVAL(j))))

S1

S1 is REAL array, dimension (max(min(MVAL(j),NVAL(j))))

S2

S2 is REAL array, dimension (max(min(MVAL(j),NVAL(j))))

X

X is COMPLEX array, dimension (LDX,NRHS)

LDX

LDX is INTEGER. The leading dimension of the arrays X, Y, and Z. LDX >= max(1,MMAX).

Y

Y is COMPLEX array, dimension (LDX,NRHS)

Z

Z is COMPLEX array, dimension (LDX,NRHS)

Q

Q is COMPLEX array, dimension (LDQ,MMAX)

LDQ

LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,MMAX).

PT

  • PT — COMPLEX array, dimension (LDPT,NMAX) 💅
  • LDPT — INTEGER. The leading dimension of the arrays PT, U, and V. It has to be LDPT >= max(1, max(min(MVAL(j),NVAL(j)))). ✨
  • U — COMPLEX array, dimension (LDPT,max(min(MVAL(j),NVAL(j)))) 🦋
  • VT — COMPLEX array, dimension (LDPT,max(min(MVAL(j),NVAL(j)))) 🦋
  • WORK — COMPLEX array, dimension (LWORK) 🫶
  • LWORK — 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)). 🔥
  • RWORK — REAL array, dimension (5*max(min(M,N))) 🌸
  • NOUT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.). 👑
  • INFO — 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: LDP < 1 or LDP < MNMAX.
    • -27: LWORK too small.
    • If CLATMR, CLATMS, CGEBRD, CUNGBR, or CBDSQR, 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. (Yes, that is what it says — don't ask me 💀)
  • 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 (the source says four, bestie — here is the list it gives 👀):

  • 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 411 of file cchkbd.f. 💖

🎀 Author

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💖 That's the whole page, bestie — source/version/date footer vibe: LAPACK Version 3.12.0, file TESTING/EIG/cchkbd.f(3). Periodt. 🎀

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