CCHKST(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 CCHKST(3) — Library Functions Manual 💖

bestie, the manual name is Library Functions Manual, and the short header path is /home/abui...IG/cchkst.f(3). buckle up, this is the test-driver page and it is long.

🌸 NAME 🌸

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

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine CCHKST (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)
           CCHKST

CCHKST is the subroutine name. iconic, no notes.

🫶 Function/Subroutine Documentation 🫶

here’s the full typed signature, exactly as the page gives it:

subroutine CCHKST (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)
       CCHKST

🌟 Purpose 🌟

okay so CCHKST checks the Hermitian eigenvalue problem routines. here’s the whole vibe, routine by routine:

  • CCHKST checks the Hermitian eigenvalue problem routines.
  • 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; CCHKST 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 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 ('I' option). It may also update an input unitary matrix, usually the output from CHETRD/CUNGTR or CHPTRD/CUPGTR ('V' option). It may also just compute eigenvalues ('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 ('I' option). CSTEMR uses the Relatively Robust Representation whenever possible.

when CCHKST is called, a number of matrix () and a number of matrix 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:

  • (1) | A - V S V* | / ( |A| n ulp ) — CHETRD( UPLO='U', ... )
  • (2) | I - UV* | / ( n ulp ) — CUNGTR( UPLO='U', ... )
  • (3) | A - V S V* | / ( |A| n ulp ) — CHETRD( UPLO='L', ... )
  • (4) | I - UV* | / ( n ulp ) — CUNGTR( UPLO='L', ... )
  • (5-8) Same as 1-4, but for CHPTRD and CUPGTR.
  • (9) | S - Z D Z* | / ( |S| n ulp ) — CSTEQR('V',...)
  • (10) | I - ZZ* | / ( n ulp ) — CSTEQR('V',...)
  • (11) | D1 - D2 | / ( |D1| ulp ) — CSTEQR('N',...)
  • (12) | D1 - D3 | / ( |D1| ulp ) — SSTERF
  • (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,

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

When S is also diagonally dominant by the factor gamma < 1,

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

and the routine is SSTEBZ( 'I', 'E', ... ).

  • (20) | S - Y WA1 Y* | / ( |S| n ulp ) — SSTEBZ, CSTEIN
  • (21) | I - Y Y* | / ( n ulp ) — SSTEBZ, CSTEIN
  • (22) | S - Z D Z* | / ( |S| n ulp ) — CSTEDC('I')
  • (23) | I - ZZ* | / ( n ulp ) — CSTEDC('I')
  • (24) | S - Z D Z* | / ( |S| n ulp ) — CSTEDC('V')
  • (25) | I - ZZ* | / ( n ulp ) — CSTEDC('V')
  • (26) | D1 - D2 | / ( |D1| ulp ) — CSTEDC('V') and CSTEDC('N')
Test 27 is disabled at the moment because CSTEMR does not guarantee high relatvie accuracy.
  • (27) max | D6(i) - WR(i) | / ( |D6(i)| omega ) , over i; omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4 — CSTEMR('V', 'A')
  • (28) max | D6(i) - WR(i) | / ( |D6(i)| omega ) , over i; omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4 — CSTEMR('V', 'I')
Tests 29 through 34 are disable at present because CSTEMR does not handle partial spectrum requests.
  • (29) | S - Z D Z* | / ( |S| n ulp ) — CSTEMR('V', 'I')
  • (30) | I - ZZ* | / ( n ulp ) — CSTEMR('V', 'I')
  • (31) the same min/max formula:
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

and it’s CSTEMR('N', 'I') vs. CSTEMR('V', 'I').

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

and it’s CSTEMR('N', 'V') vs. CSTEMR('V', 'V').

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

and it’s CSTEMR('N', 'A') vs. CSTEMR('V', 'A').

The are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type 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 ✨

Okay bestie, this is the big one — every single arg that CCHKST takes, and there are so many. Buckle in 💅

  • NSIZES — NSIZES is INTEGER. It's the number of sizes of matrices to use. If it's zero, CCHKST does literally nothing (slay, a no-op). It must be at least zero.
  • NN — NN is INTEGER array, dimension (NSIZES). It's the array holding the sizes used for the matrices. Zero values get skipped, no cap. And the values must be at least zero.
  • NTYPES — NTYPES is INTEGER. It's the number of elements in DOTYPE. If it's zero, CCHKST does nothing at all, periodt. It must be at least zero. But here's the cute special case: if it's MAXTYP+1 and NSIZES is 1, then an extra type, MAXTYP+1, is defined — and that type means "just use whatever matrix is already in A". It's only useful when 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 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. And if NTYPES is larger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) get straight-up ignored 🙈.
  • 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 they're not, they'll be reduced mod 4096. Also — important — ISEED(4) must be odd. The generator uses a linear congruential sequence limited to small integers, so it should produce machine independent random numbers (iconic). The values of ISEED are changed on exit, and you can use them in the next call to CCHKST to continue the same random number sequence 🔁.
  • THRESH — THRESH is REAL. A test will count as … if the … computed as described above exceeds THRESH (the original text has blanks right there, so we're keeping it mysterious 💀). 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 — NOUNIT is INTEGER. It's the FORTRAN unit number for printing out error messages (like, if a routine returns IINFO not equal to 0).
  • A — A is COMPLEX array of dimension ( LDA , max(NN) ). It holds the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
  • LDA — LDA is INTEGER. It's the leading dimension of A. Must be at least 1 and at least max( NN ).
  • AP — AP is COMPLEX array of dimension ( max(NN)*max(NN+1)/2 ). It's the matrix A stored in packed format.
  • SD — SD is REAL array of dimension ( max(NN) ). It's the diagonal of the tridiagonal matrix computed by CHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.
  • SE — SE is REAL array of dimension ( max(NN) ). It's the off-diagonal of the tridiagonal matrix computed by CHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A (yes, they're a matching set, we love a duo 🫶).
  • D1 — D1 is REAL array of dimension ( max(NN) ). These are the eigenvalues of A, as computed by CSTEQR simultaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A.
  • D2 — D2 is 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 — D3 is 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 — D4 is REAL array of dimension ( max(NN) ). The eigenvalues of A, as computed by CPTEQR(V). ZPTEQR factors S as Z4 D4 Z4*. On exit, the eigenvalues in D4 correspond with the matrix in A.
  • D5 — D5 is REAL array of dimension ( max(NN) ). The eigenvalues of A, as computed by ZPTEQR(N) when Z is not computed. On exit, the eigenvalues in D4 correspond with the matrix in A (bestie, the original really does say D4 here — not a typo I'm fixing, I promise 🙈).
  • WA1 — WA1 is REAL array of dimension ( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by SSTEBZ.
  • WA2 — WA2 is 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 — WA3 is 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 — WR is DOUBLE PRECISION array of dimension ( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different options, as computed by DSTEBZ.
  • U — U is COMPLEX array of dimension ( LDU, max(NN) ). It's the unitary matrix computed by CHETRD + CUNGTR ✨.
  • LDU — LDU is INTEGER. The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ).
  • V — V is 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. Little sneak-and-restore moment, iykyk 🧚)
  • VP — VP is COMPLEX array of dimension ( max(NN)*max(NN+1)/2 ). It's the matrix V stored in packed format.
  • TAU — TAU is COMPLEX array of dimension ( max(NN) ). The Householder factors computed by CHETRD in reducing A to tridiagonal form.
  • Z — Z is COMPLEX array of dimension ( LDU, max(NN) ). The unitary matrix of eigenvectors computed by CSTEQR, CPTEQR, and CSTEIN 🌈.
  • WORK — WORK is COMPLEX array of dimension ( LWORK ).
  • LWORK — LWORK is 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 — IWORK is INTEGER array. Workspace. That's it, that's the description 🫡.
  • LIWORK — LIWORK is 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 — RWORK is REAL array.
  • LRWORK — LRWORK is INTEGER. The number of entries in LRWORK (dimension( ??? ) — the original literally leaves it as ???, we stan an honest manpage 💀).
  • RESULT — RESULT is REAL array, dimension (26). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow ⚠️.
  • INFO — and we're cutting off right at the INFO arg, bestie — the rest of it rolls into the next part 🎀

okay so INFO is an INTEGER, and here's the whole vibe check on what its values mean — bestie, write these down 💅

  • 0 — everything ran OK, we love a clean run ✨
  • -1 — NSIZES < 0 (not it 🙈)
  • -2 — some NN(j) < 0
  • -3 — NTYPES < 0
  • -5 — THRESH < 0
  • -9 — LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ) — literally the max, no cap
  • -23 — LDU < 1 or LDU < NMAX
  • -29 — LWORK too small, tragic

and if CLATMR, CLATMS, CHETRD, CUNGTR, CSTEQR, SSTERF, or CUNMC2 returns an error code, the absolute value of it is returned — so it comes back positive, iykyk.

🔍 Some Local Variables and Parameters 🔍

okay these are the supporting cast, and they all matter:

  • ZERO, ONE — real 0 and 1, the basics 🎀
  • 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
  • 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, but only three are listed — we don't ask questions, we just vibe):

  • 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 599 of file cchkst.f. — periodt.

👑 Author (Doxygen's version) 👑

Generated automatically by Doxygen for LAPACK from the source code. main character energy, honestly.

that's the whole page, bestie 🫶 this iconic manual is from LAPACK Version 3.12.0, source file /home/abui...IG/cchkst.f(3) — slay and goodbye 💅✨

📦 where this page comes from