C H A P T E R 7 - Interval BLAS Routines

C H A P T E R 7

Interval BLAS Routines

Introduction

This chapter provides a brief overview of an interval Fortran 95 version of the basic linear algebra subroutine (BLAS) library. The interval BLAS version is referred to as the IBLAS library. For a more complete description of the IBLAS library routines, see the white paper Interval Version of the Basic Linear Algebra Subprograms (IBLAS).

For information on the Fortran 95 interfaces and types of arguments used in each IBLAS routine, see the section 3P man pages for the individual routines. For example, to display the man page for the SFFTC routine, type man -s 3P sfftc. Routine names must be lowercase.

For more information on the non-interval version of the BLAS library, see the document Basic Linear Algebra Subprogram Technical (BLAST) Forum Standard, available at http://www.netlib.org/blas/blast-forum/.

Note - For the Sun Studio Fortran 95 IBLAS routines, information contained in the Interval Version of the Basic Linear Algebra Subprograms (IBLAS) white paper supersedes interval information contained in the Basic Linear Algebra Subprogram Technical (BLAST) Forum Standard document that is available from NetLib.

Intervals

Intervals have a dual identity as intervals of numbers and as sets of numbers. The empty interval contains no members and is the same as the empty set in the theory of sets. In computer input and output, the empty interval is denoted [empty]. For more information on intrinsic Fortran 95 compiler support for interval data types, see the Fortran 95 Interval Arithmetic Programming Reference and the interval white papers referenced therein.

IBLAS Routine Names

This section summarizes IBLAS naming conventions derived from the BLAS specification. Language Bindings contains a list of IBLAS routine names organized into the following groups. For the corresponding detailed Fortran language bindings, see the IBLAS man pages or the IBLAS white paper.

As in the BLAS, mathematical operations and routines are grouped into:

Vector Operations Tables, listed in TABLE 7-2 through TABLE 7-4.

Matrix-Vector Operations Table, listed in TABLE 7-5.

Matrix Operations Tables, listed in TABLE 7-6 through TABLE 7-8.

New interval-specific routines are grouped into:

Set Operations on Vectors, listed in TABLE 7-9.

Set Operations on Matrices, listed in TABLE 7-10.

Utility Functions of Vectors, listed in TABLE 7-11.

Utility Functions of Matrices, listed in TABLE 7-12.

Naming Conventions

Except that the suffix _I or _i is added, IBLAS routines are named the same as the corresponding BLAS routines described in (ref BLAST Standard). IBLAS routine names have the same prefixes as the BLAS routines. Routines with prefixes identify the matrix type. TABLE 7-1 lists the IBLAS prefixes and matrix types.

TABLE 7-1 IBLAS Prefixes and Matrix Types
Prefix	Matrix Type
`GE`	General
`GB`	General Banded
`SY`	Symmetric
`SB`	Symmetric Banded
`SP`	Symmetric Packed
`TR`	Triangular
`TB`	Triangular Banded
`TP`	Triangular Packed

As in the BLAS, sparse or complex interval matrices are not treated.

A number of interval-specific, set, and utility IBLAS routines are given new BLAS-style names. See TABLE 7-9 through TABLE 7-12.

Fortran Interface

The IBLAS Fortran bindings are implemented in a module. Its interface block defines the default interval data type to be TYPE(INTERVAL).

Interval BLAS routines are consistent with regard to:

Generic interfaces

Precision

Rank

Assumed-shape arrays

Derived types

Operator arguments.

Error handling is described in the Basic Linear Algebra Subprogram Technical (BLAST) Forum Standard and in the IBLAS white paper.

Numeric error handling is not required because exceptions are not possible in the closed interval system implemented in the Sun Studio f95 compiler. Argument inconsistency errors are handled as described in IBLAS white paper, the IBLAS man pages, and the BLAST standard.

In general, actual argument shape inconsistencies cause IBLAS routines to return the largest impossible value of -1 for integer indices, a default NaN for REALvalues, and the interval R star =[ - infinity ,+] for computed intervals. The normal BLAS error handling mechanism is also used to communicate actual-parameter errors.

Binding Format

Each interface is summarized as a SUBROUTINE or FUNCTION statement, in which all the required and optional arguments appear. Optional arguments are grouped in square brackets after the required arguments. Binding format is illustrated with the Scaled Vector Sum Update (AXPBY_I) routine.

SUBROUTINE axpby_i( x, y [, alpha] [,beta] )

TYPE(INTERVAL) (<wp>), INTENT (IN) :: x (:)

TYPE(INTERVAL) (<wp>), INTENT (INOUT) :: y (:)

TYPE(INTERVAL) (<wp>), INTENT (IN), OPTIONAL :: alpha, beta

Because generic interfaces are used, the working precision, denoted <wp> is implicitly defined by the following actual arguments:

<wp> ::= KIND(4) | KIND(8) | KIND(16)

Variables in IBLAS routines are INTEGER, REAL, or TYPE(INTERVAL). See the IBLAS man pages or the IBLAS white pager for individual routine bindings.

Language Bindings

This section is a brief overview of the IBLAS Fortran routine names and their function. With the one exception of the CANCEL routines, which perform the same operation as the .DSUB. operator in f95, vector and set reductions and operations are the same as in the BLAS. The CANCEL routines and all the vector and matrix set operations and utilities are interval-specific. For interval-specific routines, the f95 equivalent scalar routines are also shown in TABLE 7-3 and TABLE 7-9 through TABLE 7-12. For clarity, lowercase and uppercase Fortran variable names are used to distinguish point from interval types. See TABLE A-11 for an alphabetical list of all the IBLAS routines.

TABLE 7-2 Vector Reductions
Name	Function
`DOT_I`	Dot Product
`NORM_I`	Vector Norms
`SUM_I`	Sum
`AMIN_VAL_I`	Minimum Absolute Value and Location
`AMAX_VAL_I`	Maximum Absolute Value and Location
`SUMSQ_I`	Scaled Sum of Squares and Update

TABLE 7-3 Add or Cancel Vectors
Name	Operation	`f95` Equivalent
`RSCALE_I`	Reciprocally Scale Vector
`AXPBY_I`	Add Scaled Vectors and Update
`WAXPBY_I`	Add Scaled Vectors
`CANCEL_I`	Cancel Scaled Vectors and Update	`Y = aX .DSUB. bY`
`WCANCEL_I`	Cancel Scaled Vectors	`W = aX .DSUB. bY`
`SUMSQ_I`	Scaled Sum of Squares and Update

TABLE 7-4 Vector Movements
Name	Operation
`COPY_I`	Vector Copy
`SWAP_I`	Vector Swap
`PERMUTE_I`	Permute Vector and Update

TABLE 7-5 Matrix-Vector Operations
Name	Operation
`{GE,GB}MV_I`	General Matrix-Vector Product and Update
`{SY,SB,SP}MV_I`	Symmetric Matrix-Vector Product and Update
`{TR,TB,TP}MV_I`	Triangular Matrix-Vector Product and Update
`{TR,TB,TP}SV_I`	Triangular Matrix Solve and Update
`GER_I`	General-Matrix Rank-One Update
`{SY,SP}R_I`	Symmetric-Matrix Rank-One Update

TABLE 7-6 O( n ² ) Matrix Operations
Name	Operation
{`GE`, `GB`, `SY`, `SB`, `SP`, `TR`, `TB`, `TP`}`_NORM_I`	Matrix Norms
{`GE`, `GB`}`_DIAG_SCALE_I`	Scale General Matrix Rows or Columns and Update
{`GE`, `GB`}`_LRSCALE_I`	Scale General Matrix Rows and Columns and Update
{`SY`, `SB`, `SP`}`_LRSCALE_I`	Scale Symmetric Matrix Rows and Columns and Update
{`GE`, `GB`, `SY`, `SB`, `SP`, `TR`, `TB`, `TP`}`_ACC_I`	Add Scaled Matrices and Update
{`GE`, `GB`, `SY`, `SB`, `SP`, `TR`, `TB`, `TP`}`_ADD_I`	Add Scaled Matrices

TABLE 7-7 O( n ³ ) Matrix Operations
Name	Operation
`GEMM_I`	General Matrix-Matrix Product and Update
`SYMM_I`	Symmetric-General Matrix-Matrix Product and Update
`TRMM_I`	Triangular-General Matrix-Matrix Product and Update
`TRSM_I`	Triangular Matrix Solve

TABLE 7-8 Matrix Movements
Name	Operation
`{GE,GB,SY,SB,SP,TR,TB,TP}_COPY_I`	Copy Matrix
`GE_TRANS_I`	Transpose Matrix
`GE_PERMUTE_I`	Permute Matrix

TABLE 7-9 Vector Set Operations
Name	Operation	`f95` Equivalent
`ENCLOSEV_I`	Enclose Vector Test	`X.SB.Y`
`INTERIORV_I`	Vector Interior Test	`X.INT.Y`
`DISJOINTV_I`	Disjoint Vector Test	`X.DJ.Y`
`INTERSECTV_I`	Intersect Vectors and Update	`Y = X.IX.Y`
`WINTERSECTV_I`	Intersect Vectors	`W = X.IX.Y`
`HULLV_I`	Hull of Vectors and Update	`Y = X.IH.Y`
`WHULLV_I`	Hull of Vectors	`W = X.IH.Y`

TABLE 7-10 Matrix Set Operations
Prefix	Name	Operation	`f95` Equivalent
	`ENCLOSEM_I`	Enclose Matrix Test	`A.SB.B`
	`INTERIORM_I`	Matrix Interior Test	`A.INT.B`
	`DISJOINTM_I`	Disjoint Matrix Test	`A.DJ.B`
`{GE`, `GB`, `SY`, `SB`, `SP`, `TR`, `TB`, `TP`}_	`INTERSECTM_I`	Intersect Matrices and Update	`B = X.IX.B`
	`WINTERSECTM_I`	Intersect Matrices	`W = X.IX.B`
	`HULLM_I`	Hull of Matrices and Update	`B = X.IH.B`
	`WHULLM_I`	Hull of Matrices	`W = X.IH.B`
^{Note: Prefix depends upon matrix type and applies to all routine names in this table.}

TABLE 7-11 Vector Utilities
Name	Operation	`f95` Equivalent
`EMPTYV_I`	Empty Vector Element Test and Location	`ISEMPTY(X)`
`INFV_I`	Vector Infimum	`v = INF(X)`
`SUPV_I`	Vector Supremum	`v = SUP(X)`
`MIDV_I`	Vector Midpoint	`v = MID(X)`
`WIDTHV_I`	Vector Width	`v = WID(X)`
`INTERVALV_I`	Vector Type Conversion to Interval	`X = INTERVAL(u,v)`

TABLE 7-12 Matrix Utilities
Prefix	Name	Operation	`f95` Equivalent
	`EMPTYM_I`	Empty Matrix Element Test and Location	`ISEMPTY(A)`
	`INFM_I`	Matrix Infimum	`c = INF(A)`
`{GE`, `GB`, `SY`, `SB`, `SP`, `TR`, `TB`, `TP`}_	`SUPM_I`	Matrix Supremum	`c = SUP(A)`
	`MIDM_I`	Matrix Midpoint	`c = MID(A)`
	`WIDTHM_I`	Matrix Width	`c = WID(A)`
	`INTERVALM_I`	Matrix Type Conversion to Interval	`A = INTERVAL(b,c)`
^{Note: Prefix depends upon matrix type and applies to all routine names in this table.}

References

The following white paper is available online. See the Interval Arithmetic readme for the location of this file.

"Interval Version of the Basic Linear Algebra Subprograms Standard (IBLAS)," derived by G.W. Walster from the draft INTERVAL BLAS Chapter 5 prepared by Chenyi Hu, et. al., to be included in the Basic Linear Algebra Subprogram Technical (BLAST) Forum Standard.