CCHKST2STG(3)

📦
version 3.12.1 · 2 versions
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🌸 TESTING/EIG/cchkst2stg.f — Library Functions Manual 🌸

okay so bestie, this page is for TESTING/EIG/cchkst2stg.f, the manual-page title is TESTING/EIG/cchkst2stg.f(3), and the manual name is Library Functions Manual. we're in the right place, no cap. 🦋

💖 NAME 💖

TESTING/EIG/cchkst2stg.f

that's the name. periodt. ✨

📜 SYNOPSIS 📜

Functions/Subroutines

subroutine cchkst2stg (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, ap, sd, se, d1, d2, d3, d4, d5, wa1, wa2, wa3, wr, u, ldu, v, vp, tau, z, work, lwork, rwork, lrwork, iwork, liwork, result, info)

CCHKST2STG

🔍 Function/Subroutine Documentation 🔍

subroutine cchkst2stg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) ap, real, dimension( * ) sd, real, dimension( * ) se, real, dimension( * ) d1, real, dimension( * ) d2, real, dimension( * ) d3, real, dimension( * ) d4, real, dimension( * ) d5, real, dimension( * ) wa1, real, dimension( * ) wa2, real, dimension( * ) wa3, real, dimension( * ) wr, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldu, * ) v, complex, dimension( * ) vp, complex, dimension( * ) tau, complex, dimension( ldu, * ) z, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, real, dimension( * ) result, integer info)

CCHKST2STG

🎯 Purpose: 🎯

okay so what is CCHKST2STG actually doing all day?? bestie, it's the vibe check for the Hermitian eigenvalue problem routines, but specifically the ones using the 2-stage reduction techniques ✨.

Real talk before we go any further though: since generating Q (or the vectors at all, honestly) isn't available in this release, we can't do the full vector comparison. So we only compare the eigenvalue results from the 2-stage path against the ones we trust as reference — the ones from the standard 1-stage reduction CHETRD. Here's the whole plan, no cap: we call the standard CHETRD and compute D1 using DSTEQR, then we call the 2-stage CHETRD_2STAGE with Upper and Lower, and we compute D2 and D3 using DSTEQR, and then we replaced tests 3 and 4 by tests 11 and 12. Tests 1 and 2 stay right where they are so we can verify the 1-stage results are OK and can be trusted 🫶.

this testing routine will converge to the CCHKST in the next release when vectors and generation of Q will be implemented. iykyk 👑

here's the cast of characters, and yes they all ate 💅

  • CHETRD factors A as U S U*, where * means conjugate transpose, S is real symmetric tridiagonal, and U is unitary. CHETRD can use either just the lower or just the upper triangle of A; CCHKST2STG checks both cases. U is represented as a product of Householder transformations, whose vectors are stored in the first n-1 columns of V, and whose scale factors are in TAU.
  • CHPTRD does the same as CHETRD, except that A and V are stored in packed format.
  • CUNGTR constructs the matrix U from the contents of V and TAU.
  • CUPGTR constructs the matrix U from the contents of VP and TAU.
  • CSTEQR factors S as Z D1 Z*, where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal. D2 is the matrix of eigenvalues computed when Z is not computed.
  • SSTERF computes D3, the matrix of eigenvalues, by the PWK method, which does not yield eigenvectors.
  • CPTEQR factors S as Z4 D4 Z4*, for a Hermitian positive definite tridiagonal matrix. D5 is the matrix of eigenvalues computed when Z is not computed.
  • SSTEBZ computes selected eigenvalues. WA1, WA2, and WA3 will denote eigenvalues computed to high absolute accuracy, with different range options. WR will denote eigenvalues computed to high relative accuracy.
  • CSTEIN computes Y, the eigenvectors of S, given the eigenvalues.
  • CSTEDC factors S as Z D1 Z*, where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal (the 'I' option). It may also update an input unitary matrix, usually the output from CHETRD/CUNGTR or CHPTRD/CUPGTR (the 'V' option). It may also just compute eigenvalues (the 'N' option).
  • CSTEMR factors S as Z D1 Z*, where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal (the 'I' option). CSTEMR uses the Relatively Robust Representation whenever possible.

so when it gets called… 🎀

When CCHKST2STG is called, a number of matrix sizes and a number of matrix types are specified. For each size and each type of matrix, one matrix will be generated and used to test the Hermitian eigenroutines. For each matrix, a number of tests will be performed 👑

  • Test 1 — | A - V S V* | / ( |A| n ulp ) via CHETRD( UPLO='U', ... )
  • Test 2 — | I - UV* | / ( n ulp ) via CUNGTR( UPLO='U', ... )
  • Test 3 — | A - V S V* | / ( |A| n ulp ) via CHETRD( UPLO='L', ... ), replaced by | D1 - D2 | / ( |D1| ulp ) where D1 is the eigenvalue matrix computed using S and D2 is the eigenvalue matrix computed using S_2stage, the output of CHETRD_2STAGE(, ,....). D1 and D2 are computed via DSTEQR('N',...)
  • Test 4 — | I - UV* | / ( n ulp ) via CUNGTR( UPLO='L', ... ), replaced by | D1 - D3 | / ( |D1| ulp ) where D1 is the eigenvalue matrix computed using S and D3 is the eigenvalue matrix computed using S_2stage, the output of CHETRD_2STAGE(, ,....). D1 and D3 are computed via DSTEQR('N',...)
  • Tests 5-8 — same as 1-4, but for CHPTRD and CUPGTR.
  • Test 9 — | S - Z D Z* | / ( |S| n ulp ) via CSTEQR('V',...)
  • Test 10 — | I - ZZ* | / ( n ulp ) via CSTEQR('V',...)
  • Test 11 — | D1 - D2 | / ( |D1| ulp ) via CSTEQR('N',...)
  • Test 12 — | D1 - D3 | / ( |D1| ulp ) via SSTERF
  • Test 13 — 0 if the true eigenvalues (computed by sturm count) of S are within THRESH of those in D1. 2*THRESH if they are not. (Tested using SSTECH)

for S positive definite 🦋

  • Test 14 — | S - Z4 D4 Z4* | / ( |S| n ulp ) via CPTEQR('V',...)
  • Test 15 — | I - Z4 Z4* | / ( n ulp ) via CPTEQR('V',...)
  • Test 16 — | D4 - D5 | / ( 100 |D4| ulp ) via CPTEQR('N',...)

when S is also diagonally dominant by the factor gamma < 1 🌸

  • Test 17 — max | D4(i) - WR(i) | / ( |D4(i)| omega ) over i, where omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4, via SSTEBZ( 'A', 'E', ...)
  • Test 18 — | WA1 - D3 | / ( |D3| ulp ) via SSTEBZ( 'A', 'E', ...)
  • Test 19 —
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

via SSTEBZ( 'I', 'E', ...)

  • Test 20 — | S - Y WA1 Y* | / ( |S| n ulp ) via SSTEBZ, CSTEIN
  • Test 21 — | I - Y Y* | / ( n ulp ) via SSTEBZ, CSTEIN
  • Test 22 — | S - Z D Z* | / ( |S| n ulp ) via CSTEDC('I')
  • Test 23 — | I - ZZ* | / ( n ulp ) via CSTEDC('I')
  • Test 24 — | S - Z D Z* | / ( |S| n ulp ) via CSTEDC('V')
  • Test 25 — | I - ZZ* | / ( n ulp ) via CSTEDC('V')
  • Test 26 — | D1 - D2 | / ( |D1| ulp ) via CSTEDC('V') and CSTEDC('N')
okay so test 27 is disabled at the moment because CSTEMR does not guarantee high relative accuracy 😭 don't come for me, it's literally in the source.
  • Test 27 — max | D6(i) - WR(i) | / ( |D6(i)| omega ) over i, where omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4, via CSTEMR('V', 'A')
  • Test 28 — max | D6(i) - WR(i) | / ( |D6(i)| omega ) over i, where omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4, via CSTEMR('V', 'I')
and tests 29 through 34 are disabled at present too, because CSTEMR does not handle partial spectrum requests 💀
  • Test 29 — | S - Z D Z* | / ( |S| n ulp ) via CSTEMR('V', 'I')
  • Test 30 — | I - ZZ* | / ( n ulp ) via CSTEMR('V', 'I')
  • Test 31 —
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

CSTEMR('N', 'I') vs. CSTEMR('V', 'I')

  • Test 32 — | S - Z D Z* | / ( |S| n ulp ) via CSTEMR('V', 'V')
  • Test 33 — | I - ZZ* | / ( n ulp ) via CSTEMR('V', 'V')
  • Test 34 —
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

CSTEMR('N', 'V') vs. CSTEMR('V', 'V')

  • Test 35 — | S - Z D Z* | / ( |S| n ulp ) via CSTEMR('V', 'A')
  • Test 36 — | I - ZZ* | / ( n ulp ) via CSTEMR('V', 'A')
  • Test 37 —
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

CSTEMR('N', 'A') vs. CSTEMR('V', 'A')

the sizes and the types 🧚

The sizes are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The types are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type j will be generated. Currently, the list of possible types is:

  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 (4), but multiplied by SQRT( overflow threshold )
  7. Same as (4), but multiplied by SQRT( underflow threshold )
  8. A matrix of the form U* D U, where U is unitary and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal.
  9. A matrix of the form U* D U, where U is unitary and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal.
  10. A matrix of the form U* D U, where U is unitary 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. Hermitian 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. Same as (8), but diagonal elements are all positive.
  17. Same as (9), but diagonal elements are all positive.
  18. Same as (10), but diagonal elements are all positive.
  19. Same as (16), but multiplied by SQRT( overflow threshold )
  20. Same as (16), but multiplied by SQRT( underflow threshold )
  21. A diagonally dominant tridiagonal matrix with geometrically spaced diagonal entries 1, ..., ULP.

🎀 Parameters 🎀

  • NSIZES — INTEGER The number of sizes of matrices to use. If it's zero, CCHKST2STG literally does nothing (iconic king of "no thoughts, head empty"). It must be at least zero.
  • NN — INTEGER array, dimension (NSIZES) An array holding all the sizes you want the matrices to be. Zero values get skipped. The values must be at least zero.
  • NTYPES — INTEGER The number of elements in DOTYPE. If it's zero, CCHKST2STG does nothing. It must be at least zero. And here's the spicy bit: if it equals MAXTYP+1 and NSIZES is 1, then one extra type — MAXTYP+1 — gets defined, and that type means "just use whatever matrix is already sitting in A". This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. 🫶
  • DOTYPE — LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j gets 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 bigger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) are just ignored, no cap.
  • ISEED — 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 they're not, they get reduced mod 4096. Also ISEED(4) must be odd (non-negotiable, it's the vibe check). The random number generator uses a linear congruential sequence limited to small integers, so it should produce machine independent random numbers. The values of ISEED change on exit, and you can reuse them in the next call to CCHKST2STG to continue the same random number sequence. ✨
  • THRESH — REAL A test will count as ␣ if the ␣, computed as described above, exceeds THRESH. (Yeah bestie, the original text literally has words missing there — not me, I promise 🙈.) Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
  • NOUNIT — INTEGER The FORTRAN unit number for printing out error messages (like if a routine returns IINFO not equal to 0).
  • A — COMPLEX array of
dimension ( LDA , max(NN) )

Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.

  • LDA — INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ).
  • AP — COMPLEX array of
dimension( max(NN)*max(NN+1)/2 )

The matrix A stored in packed format.

  • SD — REAL array of
dimension( max(NN) )

The diagonal of the tridiagonal matrix computed by CHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.

  • SE — REAL array of
dimension( max(NN) )

The off-diagonal of the tridiagonal matrix computed by CHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.

  • D1 — REAL array of
dimension( max(NN) )

The eigenvalues of A, as computed by CSTEQR simultaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A.

  • D2 — REAL array of
dimension( max(NN) )

The eigenvalues of A, as computed by CSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A.

  • D3 — REAL array of
dimension( max(NN) )

The eigenvalues of A, as computed by SSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A.

  • D4 — REAL array of
dimension( max(NN) )

The eigenvalues of A, as computed by CPTEQR(V). CPTEQR factors S as Z4 D4 Z4*. On exit, the eigenvalues in D4 correspond with the matrix in A.

  • D5 — REAL array of
dimension( max(NN) )

The eigenvalues of A, as computed by CPTEQR(N) when Z is not computed. On exit, the eigenvalues in D4 correspond with the matrix in A. (Yes, it says D4 in the D5 description — that's genuinely what the source says, I'm not making it up 💀.)

  • WA1 — REAL array of
dimension( max(NN) )

All eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by SSTEBZ.

  • WA2 — REAL array of
dimension( max(NN) )

Selected eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by SSTEBZ. Choose random values for IL and IU, and ask for the IL-th through IU-th eigenvalues.

  • WA3 — REAL array of
dimension( max(NN) )

Selected eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by SSTEBZ. Determine the values VL and VU of the IL-th and IU-th eigenvalues and ask for all eigenvalues in this range.

  • WR — REAL array of
dimension( max(NN) )

All eigenvalues of A, computed to high absolute accuracy, with different options, as computed by SSTEBZ.

  • U — COMPLEX array of
dimension( LDU, max(NN) ).

The unitary matrix computed by CHETRD + CUNGTR.

  • LDU — INTEGER The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ).
  • V — COMPLEX array of
dimension( LDU, max(NN) ).

The Householder vectors computed by CHETRD in reducing A to tridiagonal form. The vectors computed with UPLO='U' are in the upper triangle, and the vectors computed with UPLO='L' are in the lower triangle. (As described in CHETRD, the sub- and superdiagonal are not set to 1, even though the true Householder vector has a 1 in that position. The routines that use V, such as CUNGTR, set those entries to 1 before using them, and then restore them later. Respect the restore, king 🫡.)

  • VP — COMPLEX array of
dimension( max(NN)*max(NN+1)/2 )

The matrix V stored in packed format.

  • TAU — COMPLEX array of
dimension( max(NN) )

The Householder factors computed by CHETRD in reducing A to tridiagonal form.

  • Z — COMPLEX array of
dimension( LDU, max(NN) ).

The unitary matrix of eigenvectors computed by CSTEQR, CPTEQR, and CSTEIN.

  • WORK — COMPLEX array of
dimension( LWORK )
  • LWORK — INTEGER The number of entries in WORK. This must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2

where Nmax = max( NN(j), 2 ) and lg = log base 2.

  • IWORK — INTEGER array Workspace. That's it, that's the whole description 🧚
  • LIWORK — INTEGER The number of entries in IWORK. This must be at least
6 + 6*Nmax + 5 * Nmax * lg Nmax

where Nmax = max( NN(j), 2 ) and lg = log base 2.

  • RWORK — REAL array
  • LRWORK — INTEGER The number of entries in LRWORK (dimension( ??? ) — yes, the source literally has question marks there, we love a mystery 🌸).
  • RESULT — REAL array, dimension (26) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
  • INFO

okay so INFO is an INTEGER, and here's the vibe check on what each value means:

  • 0 — everything ran OK. slay 💅
  • -1 — NSIZES < 0
  • -2 — some NN(j) < 0
  • -3 — NTYPES < 0
  • -5 — THRESH < 0
  • -9 — LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
  • -23 — LDU < 1 or LDU < NMAX.
  • -29 — LWORK too small.

If CLATMR, CLATMS, CHETRD, CUNGTR, CSTEQR, SSTERF, or CUNMC2 returns an error code, the absolute value of it is returned. periodt 💖

Some local variables and parameters, bestie:

  • 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.
  • NTESTT — the total number of tests performed so far.
  • NBLOCK — blocksize as returned by ENVIR.
  • NMAX — largest value in NN.
  • NMATS — the number of matrices generated so far.
  • NERRS — the number of tests which have exceeded THRESH so far.
  • COND, IMODE — values to be passed to the matrix generators.
  • ANORM — norm of A; passed to matrix generators.

Overflow and underflow icons:

  • OVFL, UNFL — overflow and underflow thresholds.
  • ULP, ULPINV — finest relative precision and its inverse.
  • RTOVFL, RTUNFL — square roots of the previous 2 values.

The following four arrays decode JTYPE (yes, it says four and then lists three — that's the source's business, not ours 😭):

  • 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 620 of file cchkst2stg.f.

Author 🧚

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That's the whole page, bestie! 💖 This iconic little number is LAPACK Version 3.12.0, file TESTING/EIG/cchkst2stg.f(3). Catch you in the next manpage! 🌈✨

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