Improve testing of NRM2

Add tests covering arrays with extreme numbers provided in
Anderson E. (2017)
Algorithm 978: Safe Scaling in the Level 1 BLAS
ACM Trans Math Softw 44:1--28
https://doi.org/10.1145/3061665

[D,S]NRM2 coverage improves
lines     59.1% -> 100.0%
branches  30.0% ->  90.0%.

[DZ,SC]NRM2 coverage improves
lines     67.0% -> 100.0%
branches  41.7% ->  91.7%

Author: angsch

BLAS/TESTING/cblat1.f

@@ -121,7 +121,8 @@ SUBROUTINE HEADER
       SUBROUTINE CHECK1(SFAC)
 *     .. Parameters ..
       INTEGER           NOUT
-      PARAMETER         (NOUT=6)
+      REAL              THRESH
+      PARAMETER         (NOUT=6, THRESH=10.0E0)
 *     .. Scalar Arguments ..
       REAL              SFAC
 *     .. Scalars in Common ..
@@ -141,7 +142,7 @@ SUBROUTINE CHECK1(SFAC)
       INTEGER           ICAMAX
       EXTERNAL          SCASUM, SCNRM2, ICAMAX
 *     .. External Subroutines ..
-      EXTERNAL          CSCAL, CSSCAL, CTEST, ITEST1, STEST1
+      EXTERNAL          CB1NRM2, CSCAL, CSSCAL, CTEST, ITEST1, STEST1
 *     .. Intrinsic Functions ..
       INTRINSIC         MAX
 *     .. Common blocks ..
@@ -256,6 +257,10 @@ SUBROUTINE CHECK1(SFAC)
    20       CONTINUE
             IF (ICASE.EQ.6) THEN
 *              .. SCNRM2 ..
+*              Test scaling when some entries are tiny or huge
+               CALL CB1NRM2(N,(INCX-2)*2,THRESH)
+               CALL CB1NRM2(N,INCX,THRESH)
+*              Test with hardcoded mid range entries
                CALL STEST1(SCNRM2(N,CX,INCX),STRUE2(NP1),STRUE2(NP1),
      +                     SFAC)
             ELSE IF (ICASE.EQ.7) THEN
@@ -782,3 +787,225 @@ SUBROUTINE ITEST1(ICOMP,ITRUE)
 *     End of ITEST1
 *
       END
+      SUBROUTINE CB1NRM2(N,INCX,THRESH)
+*     Compare NRM2 with a reference computation using combinations
+*     of the following values:
+*
+*     0, very small, small, ulp, 1, 1/ulp, big, very big, infinity, NaN
+*
+*     one of these values is used to initialize x(1) and x(2:N) is
+*     filled with random values from [-1,1] scaled by another of
+*     these values.
+*
+*     This routine is adapted from the test suite provided by
+*     Anderson E. (2017)
+*     Algorithm 978: Safe Scaling in the Level 1 BLAS
+*     ACM Trans Math Softw 44:1--28
+*     https://doi.org/10.1145/3061665
+*
+*     .. Scalar Arguments ..
+      INTEGER           INCX, N
+      REAL              THRESH
+*
+*  =====================================================================
+*     .. Parameters ..
+      INTEGER           NMAX, NOUT, NV
+      PARAMETER         (NMAX=20, NOUT=6, NV=10)
+      REAL              HALF, ONE, THREE, TWO, ZERO
+      PARAMETER         (HALF=0.5E+0, ONE=1.0E+0, TWO= 2.0E+0,
+     &                  THREE=3.0E+0, ZERO=0.0E+0)
+*     .. External Functions ..
+      REAL              SCNRM2
+      EXTERNAL          SCNRM2
+*     .. Intrinsic Functions ..
+      INTRINSIC         AIMAG, ABS, CMPLX, MAX, MIN, REAL, SQRT
+*     .. Model parameters ..
+      REAL              BIGNUM, SAFMAX, SAFMIN, SMLNUM, ULP
+      PARAMETER         (BIGNUM=0.1014120480E+32,
+     &                  SAFMAX=0.8507059173E+38,
+     &                  SAFMIN=0.1175494351E-37,
+     &                  SMLNUM=0.9860761315E-31,
+     &                  ULP=0.1192092896E-06)
+*     .. Local Scalars ..
+      COMPLEX           ROGUE
+      REAL              SNRM, TRAT, V0, V1, WORKSSQ, Y1, Y2,
+     &                  YMAX, YMIN, YNRM, ZNRM
+      INTEGER           I, IV, IW, IX, KS
+      LOGICAL           FIRST
+*     .. Local Arrays ..
+      COMPLEX           X(NMAX), Z(NMAX)
+      REAL              VALUES(NV), WORK(NMAX)
+*     .. Executable Statements ..
+      VALUES(1) = ZERO
+      VALUES(2) = TWO*SAFMIN
+      VALUES(3) = SMLNUM
+      VALUES(4) = ULP
+      VALUES(5) = ONE
+      VALUES(6) = ONE / ULP
+      VALUES(7) = BIGNUM
+      VALUES(8) = SAFMAX
+      VALUES(9) = SXVALS(V0,2)
+      VALUES(10) = SXVALS(V0,3)
+      ROGUE = CMPLX(1234.5678E+0,-1234.5678E+0)
+      FIRST = .TRUE.
+*
+*     Check that the arrays are large enough
+*
+      IF (N*ABS(INCX).GT.NMAX) THEN
+         WRITE (NOUT,99) "SCNRM2", NMAX, INCX, N, N*ABS(INCX)
+         RETURN
+      END IF
+*
+*     Zero-sized inputs are tested in STEST1.
+      IF (N.LE.0) THEN
+         RETURN
+      END IF
+*
+*     Generate 2*(N-1) values in (-1,1).
+*
+      KS = 2*(N-1)
+      DO I = 1, KS
+         CALL RANDOM_NUMBER(WORK(I))
+         WORK(I) = ONE - TWO*WORK(I)
+      END DO
+*
+*     Compute the sum of squares of the random values
+*     by an unscaled algorithm.
+*
+      WORKSSQ = ZERO
+      DO I = 1, KS
+         WORKSSQ = WORKSSQ + WORK(I)*WORK(I)
+      END DO
+*
+*     Construct the test vector with one known value
+*     and the rest from the random work array multiplied
+*     by a scaling factor.
+*
+      DO IV = 1, NV
+         V0 = VALUES(IV)
+         IF (ABS(V0).GT.ONE) THEN
+            V0 = V0*HALF*HALF
+         END IF
+         Z(1) = CMPLX(V0,-THREE*V0)
+         DO IW = 1, NV
+            V1 = VALUES(IW)
+            IF (ABS(V1).GT.ONE) THEN
+               V1 = (V1*HALF) / SQRT(REAL(KS+1))
+            END IF
+            DO I = 1, N-1
+               Z(I+1) = CMPLX(V1*WORK(2*I-1),V1*WORK(2*I))
+            END DO
+*
+*           Compute the expected value of the 2-norm
+*
+            Y1 = ABS(V0) * SQRT(10.0E0)
+            IF (N.GT.1) THEN
+               Y2 = ABS(V1)*SQRT(WORKSSQ)
+            ELSE
+               Y2 = ZERO
+            END IF
+            YMIN = MIN(Y1, Y2)
+            YMAX = MAX(Y1, Y2)
+*
+*           Expected value is NaN if either is NaN. The test
+*           for YMIN == YMAX avoids further computation if both
+*           are infinity.
+*
+            IF ((Y1.NE.Y1).OR.(Y2.NE.Y2)) THEN
+*              add to propagate NaN
+               YNRM = Y1 + Y2
+            ELSE IF (YMIN == YMAX) THEN
+               YNRM = SQRT(TWO)*YMAX
+            ELSE IF (YMAX == ZERO) THEN
+               YNRM = ZERO
+            ELSE
+               YNRM = YMAX*SQRT(ONE + (YMIN / YMAX)**2)
+            END IF
+*
+*           Fill the input array to SCNRM2 with steps of incx
+*
+            DO I = 1, N
+               X(I) = ROGUE
+            END DO
+            IX = 1
+            IF (INCX.LT.0) IX = 1 - (N-1)*INCX
+            DO I = 1, N
+               X(IX) = Z(I)
+               IX = IX + INCX
+            END DO
+*
+*           Call SCNRM2 to compute the 2-norm
+*
+            SNRM = SCNRM2(N,X,INCX)
+*
+*           Compare SNRM and ZNRM.  Roundoff error grows like O(n)
+*           in this implementation so we scale the test ratio accordingly.
+*
+            IF (INCX.EQ.0) THEN
+               Y1 = ABS(REAL(X(1)))
+               Y2 = ABS(AIMAG(X(1)))
+               YMIN = MIN(Y1, Y2)
+               YMAX = MAX(Y1, Y2)
+               IF ((Y1.NE.Y1).OR.(Y2.NE.Y2)) THEN
+*                 add to propagate NaN
+                  ZNRM = Y1 + Y2
+               ELSE IF (YMIN == YMAX) THEN
+                  ZNRM = SQRT(TWO)*YMAX
+               ELSE IF (YMAX == ZERO) THEN
+                  ZNRM = ZERO
+               ELSE
+                  ZNRM = YMAX * SQRT(ONE + (YMIN / YMAX)**2)
+               END IF
+               ZNRM = SQRT(REAL(n)) * ZNRM
+            ELSE
+               ZNRM = YNRM
+            END IF
+            IF ((SNRM.NE.SNRM).OR.(ZNRM.NE.ZNRM)) THEN
+               IF ((SNRM.NE.SNRM).NEQV.(ZNRM.NE.ZNRM)) THEN
+                  TRAT = ONE / ULP
+               ELSE
+                  TRAT = ZERO
+               END IF
+            ELSE IF (ZNRM == ZERO) THEN
+               TRAT = SNRM / ULP
+            ELSE
+               TRAT = (ABS(SNRM-ZNRM) / ZNRM) / (TWO*REAL(N)*ULP)
+            END IF
+            IF ((TRAT.NE.TRAT).OR.(TRAT.GE.THRESH)) THEN
+               IF (FIRST) THEN
+                  FIRST = .FALSE.
+                  WRITE(NOUT,99999)
+               END IF
+               WRITE (NOUT,98) "SCNRM2", N, INCX, IV, IW, TRAT
+            END IF
+         END DO
+      END DO
+99999 FORMAT ('                                       FAIL')
+   99 FORMAT ( ' Not enough space to test ', A6, ': NMAX = ',I6,
+     + ', INCX = ',I6,/,'   N = ',I6,', must be at least ',I6 )
+   98 FORMAT( 1X, A6, ': N=', I6,', INCX=', I4, ', IV=', I2, ', IW=',
+     +  I2, ', test=', E15.8 )
+      RETURN
+      CONTAINS
+      REAL FUNCTION SXVALS(XX,K)
+*     .. Scalar Arguments ..
+      REAL              XX
+      INTEGER           K
+*     .. Local Scalars ..
+      REAL              X, Y, YY, Z
+*     .. Intrinsic Functions ..
+      INTRINSIC         HUGE
+*     .. Executable Statements ..
+      Y = HUGE(XX)
+      Z = YY
+      IF (K.EQ.1) THEN
+         X = -Z
+      ELSE IF (K.EQ.2) THEN
+         X = Z
+      ELSE IF (K.EQ.3) THEN
+         X = Z / Z
+      END IF
+      SXVALS = X
+      RETURN
+      END
+      END