SUBROUTINE DLATTP( IMAT, UPLO, TRANS, DIAG, ISEED, N, A, B, WORK, $ 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 ) DOUBLE PRECISION A( * ), B( * ), WORK( * ) * .. * * Purpose * ======= * * DLATTP 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 (= 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 * DLATMS). Modified on exit. * * N (input) INTEGER * The order of the matrix to be generated. * * A (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N) * The right hand side vector, if IMAT > 10. * * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -k, the k-th argument had an illegal value * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, TWO, ZERO PARAMETER ( ONE = 1.0D+0, TWO = 2.0D+0, ZERO = 0.0D+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 DOUBLE PRECISION ANORM, BIGNUM, BNORM, BSCAL, C, CNDNUM, PLUS1, $ PLUS2, RA, RB, REXP, S, SFAC, SMLNUM, STAR1, $ STEMP, T, TEXP, TLEFT, TSCAL, ULP, UNFL, X, Y, $ Z * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLARND EXTERNAL LSAME, IDAMAX, DLAMCH, DLARND * .. * .. External Subroutines .. EXTERNAL DLABAD, DLARNV, DLATB4, DLATMS, DROT, DROTG, $ DSCAL * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, SIGN, SQRT * .. * .. Executable Statements .. * PATH( 1: 1 ) = 'Double precision' PATH( 2: 3 ) = 'TP' UNFL = DLAMCH( 'Safe minimum' ) ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) SMLNUM = UNFL BIGNUM = ( ONE-ULP ) / SMLNUM CALL DLABAD( 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 DLATB4 to set parameters for SLATMS. * UPPER = LSAME( UPLO, 'U' ) IF( UPPER ) THEN CALL DLATB4( PATH, IMAT, N, N, TYPE, KL, KU, ANORM, MODE, $ CNDNUM, DIST ) PACKIT = 'C' ELSE CALL DLATB4( 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 DLATMS( N, N, DIST, ISEED, TYPE, B, MODE, CNDNUM, ANORM, $ KL, KU, PACKIT, A, 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 A( JC+I-1 ) = ZERO 10 CONTINUE A( JC+J-1 ) = J JC = JC + J 20 CONTINUE ELSE JC = 1 DO 40 J = 1, N A( JC ) = J DO 30 I = J + 1, N A( 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 A( JC+I ) = ZERO 50 CONTINUE A( JC+J ) = J JC = JC + J 60 CONTINUE ELSE JC = 1 DO 80 J = 1, N A( JC ) = J DO 70 I = J + 1, N A( 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 = 0.25D0 SFAC = 0.5D0 PLUS1 = SFAC 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 = DLARND( 2, ISEED ) STAR1 = STAR1*( SFAC**REXP ) IF( REXP.LT.ZERO ) THEN STAR1 = -SFAC**( ONE-REXP ) ELSE STAR1 = SFAC**( ONE+REXP ) END IF END IF 90 CONTINUE * X = SQRT( CNDNUM ) - ONE / SQRT( CNDNUM ) IF( N.GT.2 ) THEN Y = SQRT( TWO / DBLE( 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 A( JC+1 ) = Y IF( J.GT.2 ) $ A( JC+J-1 ) = WORK( J-2 ) IF( J.GT.3 ) $ A( JC+J-2 ) = WORK( N+J-3 ) JC = JC + J 100 CONTINUE JC = JC - N A( JC+1 ) = Z DO 110 J = 2, N - 1 A( 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 A( I ) = Y 120 CONTINUE A( N ) = Z JC = N + 1 DO 130 J = 2, N - 1 A( JC+1 ) = WORK( J-1 ) IF( J.LT.N-1 ) $ A( JC+2 ) = WORK( N+J-1 ) A( 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 = A( JCNEXT+J-1 ) RB = TWO CALL DROTG( RA, RB, C, S ) * * Multiply by [ c s; -s c] on the left. * IF( N.GT.J+1 ) THEN JX = JCNEXT + J DO 140 I = J + 2, N STEMP = C*A( JX+J ) + S*A( JX+J+1 ) A( JX+J+1 ) = -S*A( JX+J ) + C*A( JX+J+1 ) A( JX+J ) = STEMP JX = JX + I 140 CONTINUE END IF * * Multiply by [-c -s; s -c] on the right. * IF( J.GT.1 ) $ CALL DROT( J-1, A( JCNEXT ), 1, A( JC ), 1, -C, -S ) * * Negate A(J,J+1). * A( JCNEXT+J-1 ) = -A( JCNEXT+J-1 ) JC = JCNEXT 150 CONTINUE ELSE JC = 1 DO 170 J = 1, N - 1 JCNEXT = JC + N - J + 1 RA = A( JC+1 ) RB = TWO CALL DROTG( RA, RB, C, S ) * * Multiply by [ c -s; s c] on the right. * IF( N.GT.J+1 ) $ CALL DROT( N-J-1, A( JCNEXT+1 ), 1, A( JC+2 ), 1, C, $ -S ) * * Multiply by [-c s; -s -c] on the left. * IF( J.GT.1 ) THEN JX = 1 DO 160 I = 1, J - 1 STEMP = -C*A( JX+J-I ) + S*A( JX+J-I+1 ) A( JX+J-I+1 ) = -S*A( JX+J-I ) - C*A( JX+J-I+1 ) A( JX+J-I ) = STEMP JX = JX + N - I + 1 160 CONTINUE END IF * * Negate A(J+1,J). * A( JC+1 ) = -A( 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 DLARNV( 2, ISEED, J, A( JC ) ) A( JC+J-1 ) = SIGN( TWO, A( JC+J-1 ) ) JC = JC + J 180 CONTINUE ELSE JC = 1 DO 190 J = 1, N CALL DLARNV( 2, ISEED, N-J+1, A( JC ) ) A( JC ) = SIGN( TWO, A( JC ) ) JC = JC + N - J + 1 190 CONTINUE END IF * * Set the right hand side so that the largest value is BIGNUM. * CALL DLARNV( 2, ISEED, N, B ) IY = IDAMAX( N, B, 1 ) BNORM = ABS( B( IY ) ) BSCAL = BIGNUM / MAX( ONE, BNORM ) CALL DSCAL( 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 DLARNV( 2, ISEED, N, B ) TSCAL = ONE / MAX( ONE, DBLE( N-1 ) ) IF( UPPER ) THEN JC = 1 DO 200 J = 1, N CALL DLARNV( 2, ISEED, J-1, A( JC ) ) CALL DSCAL( J-1, TSCAL, A( JC ), 1 ) A( JC+J-1 ) = SIGN( ONE, DLARND( 2, ISEED ) ) JC = JC + J 200 CONTINUE A( N*( N+1 ) / 2 ) = SMLNUM ELSE JC = 1 DO 210 J = 1, N CALL DLARNV( 2, ISEED, N-J, A( JC+1 ) ) CALL DSCAL( N-J, TSCAL, A( JC+1 ), 1 ) A( JC ) = SIGN( ONE, DLARND( 2, ISEED ) ) JC = JC + N - J + 1 210 CONTINUE A( 1 ) = SMLNUM 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 DLARNV( 2, ISEED, N, B ) IF( UPPER ) THEN JC = 1 DO 220 J = 1, N CALL DLARNV( 2, ISEED, J-1, A( JC ) ) A( JC+J-1 ) = SIGN( ONE, DLARND( 2, ISEED ) ) JC = JC + J 220 CONTINUE A( N*( N+1 ) / 2 ) = SMLNUM ELSE JC = 1 DO 230 J = 1, N CALL DLARNV( 2, ISEED, N-J, A( JC+1 ) ) A( JC ) = SIGN( ONE, DLARND( 2, ISEED ) ) JC = JC + N - J + 1 230 CONTINUE A( 1 ) = SMLNUM 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 A( JC+I-1 ) = ZERO 240 CONTINUE IF( JCOUNT.LE.2 ) THEN A( JC+J-1 ) = SMLNUM ELSE A( JC+J-1 ) = ONE 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 A( JC+I-J ) = ZERO 260 CONTINUE IF( JCOUNT.LE.2 ) THEN A( JC ) = SMLNUM ELSE A( JC ) = ONE 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 280 CONTINUE ELSE B( N ) = ZERO DO 290 I = 1, N - 1, 2 B( I ) = ZERO B( I+1 ) = SMLNUM 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, DBLE( N-1 ) ) TSCAL = SMLNUM**TEXP CALL DLARNV( 2, ISEED, N, B ) IF( UPPER ) THEN JC = 1 DO 310 J = 1, N DO 300 I = 1, J - 2 A( JC+I-1 ) = ZERO 300 CONTINUE IF( J.GT.1 ) $ A( JC+J-2 ) = -ONE A( JC+J-1 ) = TSCAL JC = JC + J 310 CONTINUE B( N ) = ONE ELSE JC = 1 DO 330 J = 1, N DO 320 I = J + 2, N A( JC+I-J ) = ZERO 320 CONTINUE IF( J.LT.N ) $ A( JC+1 ) = -ONE A( JC ) = TSCAL JC = JC + N - J + 1 330 CONTINUE B( 1 ) = 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 DLARNV( 2, ISEED, J, A( JC ) ) IF( J.NE.IY ) THEN A( JC+J-1 ) = SIGN( TWO, A( JC+J-1 ) ) ELSE A( JC+J-1 ) = ZERO END IF JC = JC + J 340 CONTINUE ELSE JC = 1 DO 350 J = 1, N CALL DLARNV( 2, ISEED, N-J+1, A( JC ) ) IF( J.NE.IY ) THEN A( JC ) = SIGN( TWO, A( JC ) ) ELSE A( JC ) = ZERO END IF JC = JC + N - J + 1 350 CONTINUE END IF CALL DLARNV( 2, ISEED, N, B ) CALL DSCAL( 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 A( J ) = ZERO 360 CONTINUE TEXP = ONE IF( UPPER ) THEN JC = ( N-1 )*N / 2 + 1 DO 370 J = N, 2, -2 A( JC ) = -TSCAL / DBLE( N+1 ) A( JC+J-1 ) = ONE B( J ) = TEXP*( ONE-ULP ) JC = JC - J + 1 A( JC ) = -( TSCAL / DBLE( N+1 ) ) / DBLE( N+2 ) A( JC+J-2 ) = ONE B( J-1 ) = TEXP*DBLE( N*N+N-1 ) TEXP = TEXP*TWO JC = JC - J + 2 370 CONTINUE B( 1 ) = ( DBLE( N+1 ) / DBLE( N+2 ) )*TSCAL ELSE JC = 1 DO 380 J = 1, N - 1, 2 A( JC+N-J ) = -TSCAL / DBLE( N+1 ) A( JC ) = ONE B( J ) = TEXP*( ONE-ULP ) JC = JC + N - J + 1 A( JC+N-J-1 ) = -( TSCAL / DBLE( N+1 ) ) / DBLE( N+2 ) A( JC ) = ONE B( J+1 ) = TEXP*DBLE( N*N+N-1 ) TEXP = TEXP*TWO JC = JC + N - J 380 CONTINUE B( N ) = ( DBLE( N+1 ) / DBLE( 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 DLARNV( 2, ISEED, J-1, A( JC ) ) A( JC+J-1 ) = ZERO JC = JC + J 390 CONTINUE ELSE JC = 1 DO 400 J = 1, N IF( J.LT.N ) $ CALL DLARNV( 2, ISEED, N-J, A( JC+1 ) ) A( JC ) = ZERO JC = JC + N - J + 1 400 CONTINUE END IF * * Set the right hand side so that the largest value is BIGNUM. * CALL DLARNV( 2, ISEED, N, B ) IY = IDAMAX( N, B, 1 ) BNORM = ABS( B( IY ) ) BSCAL = BIGNUM / MAX( ONE, BNORM ) CALL DSCAL( 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. * TLEFT = BIGNUM / MAX( ONE, DBLE( N-1 ) ) TSCAL = BIGNUM*( DBLE( N-1 ) / MAX( ONE, DBLE( N ) ) ) IF( UPPER ) THEN JC = 1 DO 420 J = 1, N CALL DLARNV( 2, ISEED, J, A( JC ) ) DO 410 I = 1, J A( JC+I-1 ) = SIGN( TLEFT, A( JC+I-1 ) ) + $ TSCAL*A( JC+I-1 ) 410 CONTINUE JC = JC + J 420 CONTINUE ELSE JC = 1 DO 440 J = 1, N CALL DLARNV( 2, ISEED, N-J+1, A( JC ) ) DO 430 I = J, N A( JC+I-J ) = SIGN( TLEFT, A( JC+I-J ) ) + $ TSCAL*A( JC+I-J ) 430 CONTINUE JC = JC + N - J + 1 440 CONTINUE END IF CALL DLARNV( 2, ISEED, N, B ) CALL DSCAL( 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 = A( JR-I+J ) A( JR-I+J ) = A( JL ) A( 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 = A( JL+I-J ) A( JL+I-J ) = A( JR ) A( 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 DLATTP * END