ZCHKST(3)

🌸 `zchkst.f(3)` — Library Functions Manual 🌸

okay bestie, this page is literally zchkst.f(3) from the Library Functions Manual, and it's giving major LAPACK testing energy. no cap, we're looking at the whole ZCHKST situation — keep up, iykyk 💅

🌸 NAME 🌸

This whole page is about this exact file path, periodt:

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

✨ SYNOPSIS ✨

💖 Functions/Subroutines 💖

Here's the subroutine lineup, bestie — literally one icon:

subroutine ZCHKST (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)
     ZCHKST

💅 FUNCTION/SUBROUTINE DOCUMENTATION 💅

okay so this is the full signature, and it's lowkey a lot — but we do not skip a single argument in this house 🫶:

subroutine ZCHKST (integer nsizes, integer, dimension( * ) nn, integer
     ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
     double precision thresh, integer nounit, complex*16, dimension( lda, * )
     a, integer lda, complex*16, dimension( * ) ap, double precision,
     dimension( * ) sd, double precision, dimension( * ) se, double
     precision, dimension( * ) d1, double precision, dimension( * ) d2,
     double precision, dimension( * ) d3, double precision, dimension( * )
     d4, double precision, dimension( * ) d5, double precision, dimension( *
     ) wa1, double precision, dimension( * ) wa2, double precision,
     dimension( * ) wa3, double precision, dimension( * ) wr, complex*16,
     dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldu, * ) v,
     complex*16, dimension( * ) vp, complex*16, dimension( * ) tau,
     complex*16, dimension( ldu, * ) z, complex*16, dimension( * ) work,
     integer lwork, double precision, dimension( * ) rwork, integer lrwork,
     integer, dimension( * ) iwork, integer liwork, double precision,
     dimension( * ) result, integer info)
     ZCHKST

and yes, it says ZCHKST again at the end — main character energy, honestly 👑

🎀 Purpose 🎀

Okay so picture the whole Hermitian eigenvalue pipeline as a group chat, and ZCHKST is the friend who screenshots everything and checks it against the receipts. Here's who she's checking on 👑

ZHETRD factors A as U S U*, where * means conjugate transpose, S is real symmetric tridiagonal, and U is unitary. ZHETRD can use either just the lower or just the upper triangle of A — ZCHKST checks both cases, no skipping 🙅‍♀️. 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.

ZHPTRD does the same as ZHETRD, except that A and V are stored in format. (yep, the source really does leave that word blank, iykyk 🫶)

ZUNGTR constructs the matrix U from the contents of V and TAU.

ZUPGTR constructs the matrix U from the contents of VP and TAU.

ZSTEQR 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.

DSTERF computes D3, the matrix of eigenvalues, by the PWK method, which does not yield eigenvectors.

ZPTEQR 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.

DSTEBZ 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.

ZSTEIN computes Y, the eigenvectors of S, given the eigenvalues.

ZSTEDC 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 ZHETRD/ZUNGTR or ZHPTRD/ZUPGTR (the 'V' option). It may also just compute eigenvalues (the 'N' option).

ZSTEMR 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). ZSTEMR uses the Relatively Robust Representation whenever possible. iconic behavior honestly ✨

When ZCHKST 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 — and bestie, the test roster is LONG, we love a thorough queen 💅

(1) — | A - V S V* | / ( |A| n ulp ) ZHETRD( UPLO='U', ... )
(2) — | I - UV* | / ( n ulp ) ZUNGTR( UPLO='U', ... )
(3) — | A - V S V* | / ( |A| n ulp ) ZHETRD( UPLO='L', ... )
(4) — | I - UV* | / ( n ulp ) ZUNGTR( UPLO='L', ... )
(5-8) — Same as 1-4, but for ZHPTRD and ZUPGTR.
(9) — | S - Z D Z* | / ( |S| n ulp ) ZSTEQR('V',...)
(10) — | I - ZZ* | / ( n ulp ) ZSTEQR('V',...)
(11) — | D1 - D2 | / ( |D1| ulp ) ZSTEQR('N',...)
(12) — | D1 - D3 | / ( |D1| ulp ) DSTERF
(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 DSTECH)

For S positive definite:

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

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

(17) — the accuracy check against WR, done with DSTEBZ( 'A', 'E', ...):

max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
 i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4

(18) — | WA1 - D3 | / ( |D3| ulp ) DSTEBZ( 'A', 'E', ...)
(19) — the two-way min/max match-up, done with DSTEBZ( 'I', 'E', ...):

( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

(20) — | S - Y WA1 Y* | / ( |S| n ulp ) DSTEBZ, ZSTEIN
(21) — | I - Y Y* | / ( n ulp ) DSTEBZ, ZSTEIN
(22) — | S - Z D Z* | / ( |S| n ulp ) ZSTEDC('I')
(23) — | I - ZZ* | / ( n ulp ) ZSTEDC('I')
(24) — | S - Z D Z* | / ( |S| n ulp ) ZSTEDC('V')
(25) — | I - ZZ* | / ( n ulp ) ZSTEDC('V')
(26) — | D1 - D2 | / ( |D1| ulp ) ZSTEDC('V') and ZSTEDC('N')

Test 27 is disabled at the moment because ZSTEMR does not guarantee high relatvie accuracy. 😭 (that's the source's spelling, not mine, we respect the typo)

(27) — the WR accuracy check with ZSTEMR('V', 'A'):

max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
 i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4

(28) — the WR accuracy check with ZSTEMR('V', 'I'):

max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
 i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4

Tests 29 through 34 are disable at present because ZSTEMR does not handle partial spectrum requests. 💀 (again: source's wording, we stan accuracy)

(29) — | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'I')
(30) — | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'I')
(31) — the min/max match-up, ZSTEMR('N', 'I') vs. CSTEMR('V', 'I'):

( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

(32) — | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'V')
(33) — | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'V')
(34) — the min/max match-up, ZSTEMR('N', 'V') vs. CSTEMR('V', 'V'):

( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

(35) — | S - Z D Z* | / ( |S| n ulp ) ZSTEMR('V', 'A')
(36) — | I - ZZ* | / ( n ulp ) ZSTEMR('V', 'A')
(37) — the min/max match-up, ZSTEMR('N', 'A') vs. CSTEMR('V', 'A'):

( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j

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 — buckle up, there's 21 of them 🎀:

(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 — NSIZES is INTEGER. It's the number of sizes of matrices to use. If it is zero, ZCHKST does nothing. It must be at least zero.
NN — NN is INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero.
NTYPES — NTYPES is INTEGER. The number of elements in DOTYPE. If it is zero, ZCHKST 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 matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE — and yes bestie, DOTYPE gets her own whole moment, the description picks right back up on the next line of the source 🫶
DOTYPE — LOGICAL array, dimension (NTYPES). Bestie, this is your ✨to-do list✨ of matrix types: if DOTYPE(j) is .TRUE., then for every 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) just gets ignored. No cap, they simply don't matter.

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 — important, don't skip — ISEED(4) must be odd. 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 are changed on exit, and you can use them in the next call to ZCHKST to continue the exact same random number sequence. Continuity icon behavior, honestly.

THRESH — DOUBLE PRECISION. The vibe check threshold 🚨. Fair warning, the source text here has a couple of blank spots in it, so here's the line as it literally reads:

A test will count as  if the , computed as described above, exceeds THRESH.

What's not missing: 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. And it must be at least zero, periodt.

NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages 📠 — like, for example, if a routine returns IINFO not equal to 0. That's where the drama gets reported.

A — COMPLEX*16 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 — so she's not a memory, she's the moment.

LDA — INTEGER. The leading dimension of A. It must be at least 1 and at least max( NN ) 🫶.

AP — COMPLEX*16 array of dimension( max(NN)*max(NN+1)/2 ). The matrix A stored in packed format. Compact queen 💅.

SD — DOUBLE PRECISION array of dimension( max(NN) ). The diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A ✨.

SE — DOUBLE PRECISION array of dimension( max(NN) ). The off-diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A — yes, same sentence energy, they're a matched set 💖.

D1 — DOUBLE PRECISION array of dimension( max(NN) ). The eigenvalues of A, as computed by ZSTEQR simultaneously with Z 🔥. On exit, the eigenvalues in D1 correspond with the matrix in A.

D2 — DOUBLE PRECISION array of dimension( max(NN) ). The eigenvalues of A, as computed by ZSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A. Same energy, different route 🌸.

D3 — DOUBLE PRECISION array of dimension( max(NN) ). The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A 🦋.

D4 — DOUBLE PRECISION array of dimension( max(NN) ). The eigenvalues of A, as computed by ZPTEQR(V). ZPTEQR factors S as Z4 D4 Z4* ⭐. On exit, the eigenvalues in D4 correspond with the matrix in A.

D5 — DOUBLE PRECISION 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 — and yes, the original really says D4 right there in the D5 entry 🙈. We're preserving it, not fixing it, that's the rule.

WA1 — DOUBLE PRECISION array of dimension( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by DSTEBZ 👑.

WA2 — DOUBLE PRECISION array of dimension( max(NN) ). Selected eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by DSTEBZ 🎀. Specifically: choose random values for IL and IU, and ask for the IL-th through IU-th eigenvalues.

WA3 — DOUBLE PRECISION array of dimension( max(NN) ). Selected eigenvalues of A, computed to high absolute accuracy, with different range options, as computed by DSTEBZ 💜. Specifically: determine the values VL and VU of the IL-th and IU-th eigenvalues and ask for all eigenvalues in this range.

WR — DOUBLE PRECISION array of dimension( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different options, as computed by DSTEBZ 🌈.

U — COMPLEX*16 array of dimension( LDU, max(NN) ). The unitary matrix computed by ZHETRD + ZUNGTR 💅.

LDU — INTEGER. The leading dimension of U, Z, and V — all three, one number, iconic efficiency 🫶. It must be at least 1 and at least max( NN ).

V — COMPLEX*16 array of dimension( LDU, max(NN) ). The Householder vectors computed by ZHETRD 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. And here's the little lore drop, exactly as described in ZHETRD: 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 ZUNGTR, set those entries to 1 before using them, and then restore them later. Respectful, honestly 😭.

VP — COMPLEX*16 array of dimension( max(NN)*max(NN+1)/2 ). The matrix V stored in packed format 🍓.

TAU — COMPLEX*16 array of dimension( max(NN) ). The Householder factors computed by ZHETRD in reducing A to tridiagonal form ✨.

Z — COMPLEX*16 array of dimension( LDU, max(NN) ). The unitary matrix of eigenvectors computed by ZSTEQR, ZPTEQR, and ZSTEIN 🔥.

WORK — COMPLEX*16 array of dimension( LWORK ). Workspace, bestie. It works so you don't have to 💀.

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 👑. Do the math, don't guess it.

IWORK — INTEGER array. Workspace 🌸.

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 — DOUBLE PRECISION array. That's it, that's the whole description 🎀.

LRWORK — INTEGER. The number of entries in LRWORK (the original says dimension( ??? ) and honestly, same 🙈 — that question mark trio is verbatim, we keep it).

RESULT — DOUBLE PRECISION array, dimension (26). The values computed by the tests described above 💖. These values are currently limited to 1/ulp, to avoid overflow — safety first, always.

INFO — 💀 the source cuts off right here mid-entry, so this one's a cliffhanger.

INFO is INTEGER, bestie, and it’s literally the vibe report for the whole run:

If 0, then everything ran OK. 💖
-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 ZLATMR, CLATMS, ZHETRD, ZUNGTR, ZSTEQR, DSTERF, or ZUNMC2 returns an error code, the absolute value of it is returned. That’s the rule, no cap. 🫶

🧚 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.
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:

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 zchkst.f.

🫶 Author

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