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SUBROUTINE CLATTP( IMAT, UPLO, TRANS, DIAG, ISEED, N, AP, B, WORK,
$ RWORK, INFO ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER IMAT, INFO, N * .. * .. Array Arguments .. INTEGER ISEED( 4 ) REAL RWORK( * ) COMPLEX AP( * ), B( * ), WORK( * ) * .. * * Purpose * ======= * * CLATTP generates a triangular test matrix in packed storage. * IMAT and UPLO uniquely specify the properties of the test matrix, * which is returned in the array AP. * * Arguments * ========= * * IMAT (input) INTEGER * An integer key describing which matrix to generate for this * path. * * UPLO (input) CHARACTER*1 * Specifies whether the matrix A will be upper or lower * triangular. * = 'U': Upper triangular * = 'L': Lower triangular * * TRANS (input) CHARACTER*1 * Specifies whether the matrix or its transpose will be used. * = 'N': No transpose * = 'T': Transpose * = 'C': Conjugate transpose * * DIAG (output) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. * = 'N': Non-unit triangular * = 'U': Unit triangular * * ISEED (input/output) INTEGER array, dimension (4) * The seed vector for the random number generator (used in * CLATMS). Modified on exit. * * N (input) INTEGER * The order of the matrix to be generated. * * AP (output) COMPLEX array, dimension (N*(N+1)/2) * The upper or lower triangular matrix A, packed columnwise in * a linear array. The j-th column of A is stored in the array * AP as follows: * if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j; * if UPLO = 'L', * AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n. * * B (output) COMPLEX array, dimension (N) * The right hand side vector, if IMAT > 10. * * WORK (workspace) COMPLEX array, dimension (2*N) * * RWORK (workspace) REAL array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * ===================================================================== * * .. Parameters .. REAL ONE, TWO, ZERO PARAMETER ( ONE = 1.0E+0, TWO = 2.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. LOGICAL UPPER CHARACTER DIST, PACKIT, TYPE CHARACTER*3 PATH INTEGER I, IY, J, JC, JCNEXT, JCOUNT, JJ, JL, JR, JX, $ KL, KU, MODE REAL ANORM, BIGNUM, BNORM, BSCAL, C, CNDNUM, REXP, $ SFAC, SMLNUM, T, TEXP, TLEFT, TSCAL, ULP, UNFL, $ X, Y, Z COMPLEX CTEMP, PLUS1, PLUS2, RA, RB, S, STAR1 * .. * .. External Functions .. LOGICAL LSAME INTEGER ICAMAX REAL SLAMCH COMPLEX CLARND EXTERNAL LSAME, ICAMAX, SLAMCH, CLARND * .. * .. External Subroutines .. EXTERNAL CLARNV, CLATB4, CLATMS, CROT, CROTG, CSSCAL, $ SLABAD, SLARNV * .. * .. Intrinsic Functions .. INTRINSIC ABS, CMPLX, CONJG, MAX, REAL, SQRT * .. * .. Executable Statements .. * PATH( 1: 1 ) = 'Complex precision' PATH( 2: 3 ) = 'TP' UNFL = SLAMCH( 'Safe minimum' ) ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' ) SMLNUM = UNFL BIGNUM = ( ONE-ULP ) / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) IF( ( IMAT.GE.7 .AND. IMAT.LE.10 ) .OR. IMAT.EQ.18 ) THEN DIAG = 'U' ELSE DIAG = 'N' END IF INFO = 0 * * Quick return if N.LE.0. * IF( N.LE.0 ) $ RETURN * * Call CLATB4 to set parameters for CLATMS. * UPPER = LSAME( UPLO, 'U' ) IF( UPPER ) THEN CALL CLATB4( PATH, IMAT, N, N, TYPE, KL, KU, ANORM, MODE, $ CNDNUM, DIST ) PACKIT = 'C' ELSE CALL CLATB4( PATH, -IMAT, N, N, TYPE, KL, KU, ANORM, MODE, $ CNDNUM, DIST ) PACKIT = 'R' END IF * * IMAT <= 6: Non-unit triangular matrix * IF( IMAT.LE.6 ) THEN CALL CLATMS( N, N, DIST, ISEED, TYPE, RWORK, MODE, CNDNUM, $ ANORM, KL, KU, PACKIT, AP, N, WORK, INFO ) * * IMAT > 6: Unit triangular matrix * The diagonal is deliberately set to something other than 1. * * IMAT = 7: Matrix is the identity * ELSE IF( IMAT.EQ.7 ) THEN IF( UPPER ) THEN JC = 1 DO 20 J = 1, N DO 10 I = 1, J - 1 AP( JC+I-1 ) = ZERO 10 CONTINUE AP( JC+J-1 ) = J JC = JC + J 20 CONTINUE ELSE JC = 1 DO 40 J = 1, N AP( JC ) = J DO 30 I = J + 1, N AP( JC+I-J ) = ZERO 30 CONTINUE JC = JC + N - J + 1 40 CONTINUE END IF * * IMAT > 7: Non-trivial unit triangular matrix * * Generate a unit triangular matrix T with condition CNDNUM by * forming a triangular matrix with known singular values and * filling in the zero entries with Givens rotations. * ELSE IF( IMAT.LE.10 ) THEN IF( UPPER ) THEN JC = 0 DO 60 J = 1, N DO 50 I = 1, J - 1 AP( JC+I ) = ZERO 50 CONTINUE AP( JC+J ) = J JC = JC + J 60 CONTINUE ELSE JC = 1 DO 80 J = 1, N AP( JC ) = J DO 70 I = J + 1, N AP( JC+I-J ) = ZERO 70 CONTINUE JC = JC + N - J + 1 80 CONTINUE END IF * * Since the trace of a unit triangular matrix is 1, the product * of its singular values must be 1. Let s = sqrt(CNDNUM), * x = sqrt(s) - 1/sqrt(s), y = sqrt(2/(n-2))*x, and z = x**2. * The following triangular matrix has singular values s, 1, 1, * ..., 1, 1/s: * * 1 y y y ... y y z * 1 0 0 ... 0 0 y * 1 0 ... 0 0 y * . ... . . . * . . . . * 1 0 y * 1 y * 1 * * To fill in the zeros, we first multiply by a matrix with small * condition number of the form * * 1 0 0 0 0 ... * 1 + * 0 0 ... * 1 + 0 0 0 * 1 + * 0 0 * 1 + 0 0 * ... * 1 + 0 * 1 0 * 1 * * Each element marked with a '*' is formed by taking the product * of the adjacent elements marked with '+'. The '*'s can be * chosen freely, and the '+'s are chosen so that the inverse of * T will have elements of the same magnitude as T. If the *'s in * both T and inv(T) have small magnitude, T is well conditioned. * The two offdiagonals of T are stored in WORK. * * The product of these two matrices has the form * * 1 y y y y y . y y z * 1 + * 0 0 . 0 0 y * 1 + 0 0 . 0 0 y * 1 + * . . . . * 1 + . . . . * . . . . . * . . . . * 1 + y * 1 y * 1 * * Now we multiply by Givens rotations, using the fact that * * [ c s ] [ 1 w ] [ -c -s ] = [ 1 -w ] * [ -s c ] [ 0 1 ] [ s -c ] [ 0 1 ] * and * [ -c -s ] [ 1 0 ] [ c s ] = [ 1 0 ] * [ s -c ] [ w 1 ] [ -s c ] [ -w 1 ] * * where c = w / sqrt(w**2+4) and s = 2 / sqrt(w**2+4). * STAR1 = 0.25*CLARND( 5, ISEED ) SFAC = 0.5 PLUS1 = SFAC*CLARND( 5, ISEED ) DO 90 J = 1, N, 2 PLUS2 = STAR1 / PLUS1 WORK( J ) = PLUS1 WORK( N+J ) = STAR1 IF( J+1.LE.N ) THEN WORK( J+1 ) = PLUS2 WORK( N+J+1 ) = ZERO PLUS1 = STAR1 / PLUS2 REXP = CLARND( 2, ISEED ) IF( REXP.LT.ZERO ) THEN STAR1 = -SFAC**( ONE-REXP )*CLARND( 5, ISEED ) ELSE STAR1 = SFAC**( ONE+REXP )*CLARND( 5, ISEED ) END IF END IF 90 CONTINUE * X = SQRT( CNDNUM ) - ONE / SQRT( CNDNUM ) IF( N.GT.2 ) THEN Y = SQRT( TWO / REAL( N-2 ) )*X ELSE Y = ZERO END IF Z = X*X * IF( UPPER ) THEN * * Set the upper triangle of A with a unit triangular matrix * of known condition number. * JC = 1 DO 100 J = 2, N AP( JC+1 ) = Y IF( J.GT.2 ) $ AP( JC+J-1 ) = WORK( J-2 ) IF( J.GT.3 ) $ AP( JC+J-2 ) = WORK( N+J-3 ) JC = JC + J 100 CONTINUE JC = JC - N AP( JC+1 ) = Z DO 110 J = 2, N - 1 AP( JC+J ) = Y 110 CONTINUE ELSE * * Set the lower triangle of A with a unit triangular matrix * of known condition number. * DO 120 I = 2, N - 1 AP( I ) = Y 120 CONTINUE AP( N ) = Z JC = N + 1 DO 130 J = 2, N - 1 AP( JC+1 ) = WORK( J-1 ) IF( J.LT.N-1 ) $ AP( JC+2 ) = WORK( N+J-1 ) AP( JC+N-J ) = Y JC = JC + N - J + 1 130 CONTINUE END IF * * Fill in the zeros using Givens rotations * IF( UPPER ) THEN JC = 1 DO 150 J = 1, N - 1 JCNEXT = JC + J RA = AP( JCNEXT+J-1 ) RB = TWO CALL CROTG( RA, RB, C, S ) * * Multiply by [ c s; -conjg(s) c] on the left. * IF( N.GT.J+1 ) THEN JX = JCNEXT + J DO 140 I = J + 2, N CTEMP = C*AP( JX+J ) + S*AP( JX+J+1 ) AP( JX+J+1 ) = -CONJG( S )*AP( JX+J ) + $ C*AP( JX+J+1 ) AP( JX+J ) = CTEMP JX = JX + I 140 CONTINUE END IF * * Multiply by [-c -s; conjg(s) -c] on the right. * IF( J.GT.1 ) $ CALL CROT( J-1, AP( JCNEXT ), 1, AP( JC ), 1, -C, -S ) * * Negate A(J,J+1). * AP( JCNEXT+J-1 ) = -AP( JCNEXT+J-1 ) JC = JCNEXT 150 CONTINUE ELSE JC = 1 DO 170 J = 1, N - 1 JCNEXT = JC + N - J + 1 RA = AP( JC+1 ) RB = TWO CALL CROTG( RA, RB, C, S ) S = CONJG( S ) * * Multiply by [ c -s; conjg(s) c] on the right. * IF( N.GT.J+1 ) $ CALL CROT( N-J-1, AP( JCNEXT+1 ), 1, AP( JC+2 ), 1, C, $ -S ) * * Multiply by [-c s; -conjg(s) -c] on the left. * IF( J.GT.1 ) THEN JX = 1 DO 160 I = 1, J - 1 CTEMP = -C*AP( JX+J-I ) + S*AP( JX+J-I+1 ) AP( JX+J-I+1 ) = -CONJG( S )*AP( JX+J-I ) - $ C*AP( JX+J-I+1 ) AP( JX+J-I ) = CTEMP JX = JX + N - I + 1 160 CONTINUE END IF * * Negate A(J+1,J). * AP( JC+1 ) = -AP( JC+1 ) JC = JCNEXT 170 CONTINUE END IF * * IMAT > 10: Pathological test cases. These triangular matrices * are badly scaled or badly conditioned, so when used in solving a * triangular system they may cause overflow in the solution vector. * ELSE IF( IMAT.EQ.11 ) THEN * * Type 11: Generate a triangular matrix with elements between * -1 and 1. Give the diagonal norm 2 to make it well-conditioned. * Make the right hand side large so that it requires scaling. * IF( UPPER ) THEN JC = 1 DO 180 J = 1, N CALL CLARNV( 4, ISEED, J-1, AP( JC ) ) AP( JC+J-1 ) = CLARND( 5, ISEED )*TWO JC = JC + J 180 CONTINUE ELSE JC = 1 DO 190 J = 1, N IF( J.LT.N ) $ CALL CLARNV( 4, ISEED, N-J, AP( JC+1 ) ) AP( JC ) = CLARND( 5, ISEED )*TWO JC = JC + N - J + 1 190 CONTINUE END IF * * Set the right hand side so that the largest value is BIGNUM. * CALL CLARNV( 2, ISEED, N, B ) IY = ICAMAX( N, B, 1 ) BNORM = ABS( B( IY ) ) BSCAL = BIGNUM / MAX( ONE, BNORM ) CALL CSSCAL( N, BSCAL, B, 1 ) * ELSE IF( IMAT.EQ.12 ) THEN * * Type 12: Make the first diagonal element in the solve small to * cause immediate overflow when dividing by T(j,j). * In type 12, the offdiagonal elements are small (CNORM(j) < 1). * CALL CLARNV( 2, ISEED, N, B ) TSCAL = ONE / MAX( ONE, REAL( N-1 ) ) IF( UPPER ) THEN JC = 1 DO 200 J = 1, N CALL CLARNV( 4, ISEED, J-1, AP( JC ) ) CALL CSSCAL( J-1, TSCAL, AP( JC ), 1 ) AP( JC+J-1 ) = CLARND( 5, ISEED ) JC = JC + J 200 CONTINUE AP( N*( N+1 ) / 2 ) = SMLNUM*AP( N*( N+1 ) / 2 ) ELSE JC = 1 DO 210 J = 1, N CALL CLARNV( 2, ISEED, N-J, AP( JC+1 ) ) CALL CSSCAL( N-J, TSCAL, AP( JC+1 ), 1 ) AP( JC ) = CLARND( 5, ISEED ) JC = JC + N - J + 1 210 CONTINUE AP( 1 ) = SMLNUM*AP( 1 ) END IF * ELSE IF( IMAT.EQ.13 ) THEN * * Type 13: Make the first diagonal element in the solve small to * cause immediate overflow when dividing by T(j,j). * In type 13, the offdiagonal elements are O(1) (CNORM(j) > 1). * CALL CLARNV( 2, ISEED, N, B ) IF( UPPER ) THEN JC = 1 DO 220 J = 1, N CALL CLARNV( 4, ISEED, J-1, AP( JC ) ) AP( JC+J-1 ) = CLARND( 5, ISEED ) JC = JC + J 220 CONTINUE AP( N*( N+1 ) / 2 ) = SMLNUM*AP( N*( N+1 ) / 2 ) ELSE JC = 1 DO 230 J = 1, N CALL CLARNV( 4, ISEED, N-J, AP( JC+1 ) ) AP( JC ) = CLARND( 5, ISEED ) JC = JC + N - J + 1 230 CONTINUE AP( 1 ) = SMLNUM*AP( 1 ) END IF * ELSE IF( IMAT.EQ.14 ) THEN * * Type 14: T is diagonal with small numbers on the diagonal to * make the growth factor underflow, but a small right hand side * chosen so that the solution does not overflow. * IF( UPPER ) THEN JCOUNT = 1 JC = ( N-1 )*N / 2 + 1 DO 250 J = N, 1, -1 DO 240 I = 1, J - 1 AP( JC+I-1 ) = ZERO 240 CONTINUE IF( JCOUNT.LE.2 ) THEN AP( JC+J-1 ) = SMLNUM*CLARND( 5, ISEED ) ELSE AP( JC+J-1 ) = CLARND( 5, ISEED ) END IF JCOUNT = JCOUNT + 1 IF( JCOUNT.GT.4 ) $ JCOUNT = 1 JC = JC - J + 1 250 CONTINUE ELSE JCOUNT = 1 JC = 1 DO 270 J = 1, N DO 260 I = J + 1, N AP( JC+I-J ) = ZERO 260 CONTINUE IF( JCOUNT.LE.2 ) THEN AP( JC ) = SMLNUM*CLARND( 5, ISEED ) ELSE AP( JC ) = CLARND( 5, ISEED ) END IF JCOUNT = JCOUNT + 1 IF( JCOUNT.GT.4 ) $ JCOUNT = 1 JC = JC + N - J + 1 270 CONTINUE END IF * * Set the right hand side alternately zero and small. * IF( UPPER ) THEN B( 1 ) = ZERO DO 280 I = N, 2, -2 B( I ) = ZERO B( I-1 ) = SMLNUM*CLARND( 5, ISEED ) 280 CONTINUE ELSE B( N ) = ZERO DO 290 I = 1, N - 1, 2 B( I ) = ZERO B( I+1 ) = SMLNUM*CLARND( 5, ISEED ) 290 CONTINUE END IF * ELSE IF( IMAT.EQ.15 ) THEN * * Type 15: Make the diagonal elements small to cause gradual * overflow when dividing by T(j,j). To control the amount of * scaling needed, the matrix is bidiagonal. * TEXP = ONE / MAX( ONE, REAL( N-1 ) ) TSCAL = SMLNUM**TEXP CALL CLARNV( 4, ISEED, N, B ) IF( UPPER ) THEN JC = 1 DO 310 J = 1, N DO 300 I = 1, J - 2 AP( JC+I-1 ) = ZERO 300 CONTINUE IF( J.GT.1 ) $ AP( JC+J-2 ) = CMPLX( -ONE, -ONE ) AP( JC+J-1 ) = TSCAL*CLARND( 5, ISEED ) JC = JC + J 310 CONTINUE B( N ) = CMPLX( ONE, ONE ) ELSE JC = 1 DO 330 J = 1, N DO 320 I = J + 2, N AP( JC+I-J ) = ZERO 320 CONTINUE IF( J.LT.N ) $ AP( JC+1 ) = CMPLX( -ONE, -ONE ) AP( JC ) = TSCAL*CLARND( 5, ISEED ) JC = JC + N - J + 1 330 CONTINUE B( 1 ) = CMPLX( ONE, ONE ) END IF * ELSE IF( IMAT.EQ.16 ) THEN * * Type 16: One zero diagonal element. * IY = N / 2 + 1 IF( UPPER ) THEN JC = 1 DO 340 J = 1, N CALL CLARNV( 4, ISEED, J, AP( JC ) ) IF( J.NE.IY ) THEN AP( JC+J-1 ) = CLARND( 5, ISEED )*TWO ELSE AP( JC+J-1 ) = ZERO END IF JC = JC + J 340 CONTINUE ELSE JC = 1 DO 350 J = 1, N CALL CLARNV( 4, ISEED, N-J+1, AP( JC ) ) IF( J.NE.IY ) THEN AP( JC ) = CLARND( 5, ISEED )*TWO ELSE AP( JC ) = ZERO END IF JC = JC + N - J + 1 350 CONTINUE END IF CALL CLARNV( 2, ISEED, N, B ) CALL CSSCAL( N, TWO, B, 1 ) * ELSE IF( IMAT.EQ.17 ) THEN * * Type 17: Make the offdiagonal elements large to cause overflow * when adding a column of T. In the non-transposed case, the * matrix is constructed to cause overflow when adding a column in * every other step. * TSCAL = UNFL / ULP TSCAL = ( ONE-ULP ) / TSCAL DO 360 J = 1, N*( N+1 ) / 2 AP( J ) = ZERO 360 CONTINUE TEXP = ONE IF( UPPER ) THEN JC = ( N-1 )*N / 2 + 1 DO 370 J = N, 2, -2 AP( JC ) = -TSCAL / REAL( N+1 ) AP( JC+J-1 ) = ONE B( J ) = TEXP*( ONE-ULP ) JC = JC - J + 1 AP( JC ) = -( TSCAL / REAL( N+1 ) ) / REAL( N+2 ) AP( JC+J-2 ) = ONE B( J-1 ) = TEXP*REAL( N*N+N-1 ) TEXP = TEXP*TWO JC = JC - J + 2 370 CONTINUE B( 1 ) = ( REAL( N+1 ) / REAL( N+2 ) )*TSCAL ELSE JC = 1 DO 380 J = 1, N - 1, 2 AP( JC+N-J ) = -TSCAL / REAL( N+1 ) AP( JC ) = ONE B( J ) = TEXP*( ONE-ULP ) JC = JC + N - J + 1 AP( JC+N-J-1 ) = -( TSCAL / REAL( N+1 ) ) / REAL( N+2 ) AP( JC ) = ONE B( J+1 ) = TEXP*REAL( N*N+N-1 ) TEXP = TEXP*TWO JC = JC + N - J 380 CONTINUE B( N ) = ( REAL( N+1 ) / REAL( N+2 ) )*TSCAL END IF * ELSE IF( IMAT.EQ.18 ) THEN * * Type 18: Generate a unit triangular matrix with elements * between -1 and 1, and make the right hand side large so that it * requires scaling. * IF( UPPER ) THEN JC = 1 DO 390 J = 1, N CALL CLARNV( 4, ISEED, J-1, AP( JC ) ) AP( JC+J-1 ) = ZERO JC = JC + J 390 CONTINUE ELSE JC = 1 DO 400 J = 1, N IF( J.LT.N ) $ CALL CLARNV( 4, ISEED, N-J, AP( JC+1 ) ) AP( JC ) = ZERO JC = JC + N - J + 1 400 CONTINUE END IF * * Set the right hand side so that the largest value is BIGNUM. * CALL CLARNV( 2, ISEED, N, B ) IY = ICAMAX( N, B, 1 ) BNORM = ABS( B( IY ) ) BSCAL = BIGNUM / MAX( ONE, BNORM ) CALL CSSCAL( N, BSCAL, B, 1 ) * ELSE IF( IMAT.EQ.19 ) THEN * * Type 19: Generate a triangular matrix with elements between * BIGNUM/(n-1) and BIGNUM so that at least one of the column * norms will exceed BIGNUM. * 1/3/91: CLATPS no longer can handle this case * TLEFT = BIGNUM / MAX( ONE, REAL( N-1 ) ) TSCAL = BIGNUM*( REAL( N-1 ) / MAX( ONE, REAL( N ) ) ) IF( UPPER ) THEN JC = 1 DO 420 J = 1, N CALL CLARNV( 5, ISEED, J, AP( JC ) ) CALL SLARNV( 1, ISEED, J, RWORK ) DO 410 I = 1, J AP( JC+I-1 ) = AP( JC+I-1 )*( TLEFT+RWORK( I )*TSCAL ) 410 CONTINUE JC = JC + J 420 CONTINUE ELSE JC = 1 DO 440 J = 1, N CALL CLARNV( 5, ISEED, N-J+1, AP( JC ) ) CALL SLARNV( 1, ISEED, N-J+1, RWORK ) DO 430 I = J, N AP( JC+I-J ) = AP( JC+I-J )* $ ( TLEFT+RWORK( I-J+1 )*TSCAL ) 430 CONTINUE JC = JC + N - J + 1 440 CONTINUE END IF CALL CLARNV( 2, ISEED, N, B ) CALL CSSCAL( N, TWO, B, 1 ) END IF * * Flip the matrix across its counter-diagonal if the transpose will * be used. * IF( .NOT.LSAME( TRANS, 'N' ) ) THEN IF( UPPER ) THEN JJ = 1 JR = N*( N+1 ) / 2 DO 460 J = 1, N / 2 JL = JJ DO 450 I = J, N - J T = AP( JR-I+J ) AP( JR-I+J ) = AP( JL ) AP( JL ) = T JL = JL + I 450 CONTINUE JJ = JJ + J + 1 JR = JR - ( N-J+1 ) 460 CONTINUE ELSE JL = 1 JJ = N*( N+1 ) / 2 DO 480 J = 1, N / 2 JR = JJ DO 470 I = J, N - J T = AP( JL+I-J ) AP( JL+I-J ) = AP( JR ) AP( JR ) = T JR = JR - I 470 CONTINUE JL = JL + N - J + 1 JJ = JJ - J - 1 480 CONTINUE END IF END IF * RETURN * * End of CLATTP * END |