The number of columns (or rows) of

A

.

On entry,

dist

specifies the type of distribution to be used to generate the random eigen-/singular values, and on the upper triangle (see

upper

).

If

dist

=

'U'

: uniform( 0, 1 )

If

dist

=

'S'

: uniform( -1, 1 )

If

dist

=

'N'

: normal( 0, 1 )

If

dist

=

'D'

: uniform on the complex disc |

z

| < 1.

Array, size 4.

On entry

iseed

specifies the seed of the random number generator. The elements should lie between 0 and 4095 inclusive, and

iseed

[3]

should be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers.

Array, size (

n

). This array is used to specify the eigenvalues of

A

.

If

mode

= 0

, then

d

is assumed to contain the eigenvalues. Otherwise they are computed according to

mode

,

cond

,

dmax

, and

rsign

and placed in

d

.

On entry

mode

describes how the eigenvalues are to be specified:

mode

= 0

means use

d

(with

ei

for

slatme

and

dlatme

) as input.

mode

= 1

sets

d

[0]

= 1

and

d

(2:

n

]

=1.0/

cond

.

mode

= 2

sets

d

[0:

n

- 2]

= 1

and

d

[

n

- 1]

=1.0/

cond

.

mode

= 3

sets

d

[

i

- 1]

=

cond

**(-(

i

-1)/(

n

-1))

.

mode

= 4

sets

d

[

i

- 1]

= 1 - (

i

-1)/(

n

-1)*(1 - 1/

cond

)

.

mode

= 5

sets

d

to random numbers in the range

( 1/

cond

, 1 )

such that their logarithms are uniformly distributed.

mode

= 6

sets

d

to random numbers from same distribution as the rest of the matrix.

mode

< 0

has the same meaning as

abs(

mode

)

, except that the order of the elements of

d

is reversed.

Thus if

mode

is between 1 and 4,

d

has entries ranging from 1 to 1/

cond

, if between -1 and -4,

d

has entries ranging from 1/

cond

to 1.

On entry, this is used as described under

mode

above. If used, it must be

≥

1.

If

mode

is not -6, 0 or 6, the contents of

d

as computed according to

mode

and

cond

are scaled by

dmax

/ max(abs(

d

[

i

- 1]

))

. Note that

dmax

needs not be positive or real: if

dmax

is negative or complex (or zero),

d

will be scaled by a negative or complex number (or zero). If

rsign

=

'F'

then the largest (absolute) eigenvalue will be equal to

dmax

.

Used by

slatme

and

dlatme

only.

Array, size (

n

).

If

mode

= 0

, and

ei

[0]

is not

' '

(space character), this array specifies which elements of

d

(on input) are real eigenvalues and which are the real and imaginary parts of a complex conjugate pair of eigenvalues. The elements of

ei

may then only have the values

'R'

and

'I'

.

If

ei

[

j

- 1]

=

'R'

and

ei

[

j

]

=

'I'

, then the

j

-th eigenvalue is cmplx(

d

[

j

- 1]

,

d

[

j

]

), and the (

j

+1)-th is the complex conjugate thereof.

If

ei

[

j

- 1]

=

ei

[

j

]

=

'R'

, then the

j

-th eigenvalue is

d

[

j

- 1]

(i.e., real).

ei

[0]

may not be

'I'

, nor may two adjacent elements of

ei

both have the value

'I'

.

If

mode

is not 0, then

ei

is ignored. If

mode

is 0 and

ei

[0]

=

' '

, then the eigenvalues will all be real.

If

mode

is not 0, 6, or -6, and

rsign

=

'T'

, then the elements of

d

, as computed according to

mode

and

cond

, are multiplied by a random sign (+1 or -1) for

slatme

and

dlatme

or by a complex number from the unit circle |

z

| = 1 for

clatme

and

zlatme

.

If

rsign

=

'F'

, the elements of

d

are not multiplied.

rsign

may only have the values

'T'

or

'F'

.

If

upper

=

'T'

, then the elements of

a

above the diagonal will be set to random numbers out of

dist

.

If

upper

=

'F'

, they will not.

upper

may only have the values

'T'

or

'F'

.

If

sim

=

'T'

, then

a

will be operated on by a "similarity transform", i.e., multiplied on the left by a matrix

X

and on the right by

X

inverse.

X

=

U

S

V

, where

U

and

V

are random unitary matrices and

S

is a (diagonal) matrix of singular values specified by

ds

,

modes

, and

conds

.

If

sim

=

'F'

, then

a

will not be transformed.

This array is used to specify the singular values of

X

, in the same way that

d

specifies the eigenvalues of

a

. If

mode

= 0

, the

ds

contains the singular values, which may not be zero.

Similar to

mode

, but for specifying the diagonal of

S

.

modes

= -6 and +6 are not allowed (since they would result in randomly ill-conditioned eigenvalues.)

Similar to

cond

, but for specifying the diagonal of

S

.

This specifies the lower bandwidth of the matrix.

kl

= 1

specifies upper Hessenberg form. If

kl

is at least

n

-1, then

A

will have full lower bandwidth.

This specifies the upper bandwidth of the matrix.

ku

= 1

specifies lower Hessenberg form.

If

ku

is at least

n

-1, then

a

will have full upper bandwidth.

If

ku

and

ku

are both at least

n

-1, then

a

will be dense. Only one of

ku

and

kl

may be less than

n

-1.

If

anorm

is not negative, then

a

is scaled by a non-negative real number to make the maximum-element-norm of

a

to be

anorm

.

Number of rows of matrix

A

.

Array, size

(3*

n

).

Workspace.

On exit, the seed is updated.

Modified if

mode

is nonzero.

Modified if

mode

is nonzero.

Array, size

lda

*

n

. On exit,

a

is the desired test matrix.

If

info

= 0

, execution is successful.

If

info

= -1

,

n

is negative .

If

info

= -2

,

dist

is an illegal string.

If

info

= -5

,

mode

is not in range -6 to 6.

If

info

= -6

,

cond

is less than 1.0, and

mode

is not -6, 0, or 6 .

If

info

= -9

,

rsign

is not

'T'

or

'F'

.

If

info

= -10

,

upper

is not

'T'

or

'F'

.

If

info

= -11

,

sim

is not

'T'

or

'F'

.

If

info

= -12

,

modes

= 0 and

ds

has a zero singular value.

If

info

= -13

,

modes

is not in the range -5 to 5.

If

info

= -14

,

modes

is nonzero and

conds

is less than 1. .

If

info

= -15

,

kl

is less than 1.

If

info

= -16

,

ku

is less than 1, or

kl

and

ku

are both less than

n

-1.

If

info

= -19

,

lda

is less than

m

.

If

info

= 1

, error return from

?latm1

(computing

d

) .

If

info

= 2

, cannot scale to

dmax

(max. eigenvalue is 0) .

If

info

= 3

, error return from

slatm1

(for

slatme

and

clatme

),

dlatm1

(for

dlatme

and

zlatme

) .

If

info

= 4

, error return from

?large

.

If

info

= 5

, zero singular value from

slatm1

(for

slatme

and

clatme

),

dlatm1

(for

dlatme

and

zlatme

).