Tim Stark wrote:

> Hello folks,

>

> Well, I found a geodesy book called "Linear Algebra, Geodesy, and

> GPS". Is that for terra mapping and nagivation programming? Does

> anyone have good reviews with that book?

>

> Thanks,

> Tim
"Linear Algebra, Geodesy, and GPS" by Gilbert Strang and Kai Borre

is an excellent book.

Linear Algebra, Geodesy, and GPS by Gilbert Strang and Kai Borre

Wellesley-Cambridge Press Box 812060 Wellesley MA 02181

fax 617 253-4358 phone 781 431-8488 email

http://www.gps-forums.net/(E-Mail Removed)
640 pages (1997) hardcover ISBN 0-9614088-6-3

http://www-math.mit.edu/~gs
The book has three closely connected parts. The first is Linear Algebra,

which is the fundamental tool in positioning calculations. The second is

Geodesy, which includes the correct weightings for observation errors and

the development of least squares. The third is GPS, which we describe in

detail at several levels : first the basic structure of the GPS system, then

the algorithms that yield accurate positions from inaccurate pseudoranges,

and finally the Kalman filtering (and "Bayes" filtering) that give superior

accuracy in postprocessing a long series of observations.

If you wanted to order this book, I only need to ask for email with the best

mailing address. The book will come with an invoice ; a check to the Press

(not credit card) is perfect. The cost including domestic postage is $60.

Orders can equally well go to Wellesley-Cambridge Press (fax 617 253 4358).

For international orders the cost with surface postage is $62. I now

recommend the new Global Priority Airmail (if available to your country)

which only adds $5 to that cost. Personal checks in US dollars or

Eurocheques in English pounds are fine. Often the easiest is cash to

the Press and it is amazingly safe ; we take all risk.

**Please let me know what you prefer and the Press will follow through.**

A large library of MATLAB files is associated with the book. They were

created by Kai Borre and are freely available on the web. We hope that these

M-files will help readers to use the more individual and experimental

processing that MATLAB offers, compared with large-scale software packages.

I will enclose the Table of Contents below. I would be very grateful

if you could make this book known to friends. You could send them this

message and/or just show them the book ! Most of all I hope you will enjoy

this presentation of a technology that is developing so fast.

With thanks and very best regards, Gilbert Strang

TABLE OF CONTENTS

Preface........................................... ........ix

The Mathematics of GPS..................................xiii

Part I Linear Algebra

1 Vectors and Matrices....................................3

1.1 Vectors........................................... ..3

1.2 Lengths and Dot Products...........................11

1.3 Planes............................................ .20

1.4 Matrices and Linear Equations......................28

2 Solving Linear Equations...............................37

2.1 The Idea of Elimination............................37

2.2 Elimination Using Matrices.........................46

2.3 Rules for Matrix Operations........................54

2.4 Inverse Matrices...................................65

2.5 Elimination = Factorization: A = LU................75

2.6 Transposes and Permutations........................87

3 Vector Spaces and Subspaces...........................101

3.1 Spaces of Vectors.................................101

3.2 The Nullspace of A: Solving Ax = 0................109

3.3 The Rank of A: Solving Ax = b.....................122

3.4 Independence, Basis, and Dimension................134

3.5 Dimensions of the Four Subspaces..................146

4 Orthogonality..................................... ....157

4.1 Orthogonality of the Four Subspaces...............157

4.2 Projections....................................... 165

4.3 Least-Squares Approximations......................174

4.4 Orthogonal Bases and Gram-Schmidt.................184

5 Determinants...................................... ....197

5.1 The Properties of Determinants....................197

5.2 Cramer's Rule, Inverses, and Volumes..............206

6 Eigenvalues and Eigenvectors..........................211

6.1 Introduction to Eigenvalues.......................211

6.2 Diagonalizing a Matrix............................221

6.3 Symmetric Matrices................................233

6.4 Positive Definite Matrices........................237

6.5 Stability and Preconditioning.....................248

7 Linear Transformations................................251

7.1 The Idea of a Linear Transformation...............251

7.2 Choice of Basis: Similarity and SVD...............258

Part II Geodesy

8 Leveling Networks.....................................275

8.1 Heights by Least Squares..........................275

8.2 Weighted Least Squares............................280

8.3 Leveling Networks and Graphs......................282

8.4 Graphs and Incidence Matrices.....................288

8.5 One-Dimensional Distance Networks.................305

9 Random Variables and Covariance Matrices..............309

9.1 The Normal Distribution and X2...................309

9.2 Mean, Variance, and Standard Deviation............319

9.3 Covariance........................................ 320

9.4 Inverse Covariances as Weights....................322

9.5 Estimation of Mean and Variance...................326

9.6 Propagation of Means and Covariances..............328

9.7 Estimating the Variance of Unit Weight............333

9.8 Confidence Ellipses...............................337

10 Nonlinear Problems....................................343

10.1 Getting Around Nonlinearity......................343

10.2 Geodetic Observation Equations...................349

10.3 Three-Dimensional Model..........................362

11 Linear Algebra for Weighted Least Squares.............369

11.1 Gram-Schmidt on A and Cholesky on A T A..........369

11.2 Cholesky's Method in the Least-Squares Setting...372

11.3 SVD: The Canonical Form for Geodesy..............375

11.4 The Condition Number.............................377

11.5 Regularly Spaced Networks........................379

11.6 Dependency on the Weights........................391

11.7 Elimination of Unknowns..........................394

11.8 Decorrelation and Weight Normalization...........400

12 Constraints for Singular Normal Equations.............405

12.1 Rank Deficient Normal Equations..................405

12.2 Representations of the Nullspace.................406

12.3 Constraining a Rank Deficient Problem............408

12.4 Linear Transformation of Random Variables........413

12.5 Similarity Transformations.......................414

12.6 Covariance Transformations.......................421

12.7 Variances at Control Points......................423

13 Problems With Explicit Solutions......................431

13.1 Free Stationing as a Similarity Transformation...431

13.2 Optimum Choice of Observation Site...............434

13.3 Station Adjustment...............................438

13.4 Fitting a Straight Line..........................441

Part III Global Positioning System (GPS)

14 Global Positioning System.............................447

14.1 Positioning by GPS...............................447

14.2 Errors in the GPS Observables....................453

14.3 Description of the System........................458

14.4 Receiver Position From Code Observations.........460

14.5 Combined Code and Phase Observations.............463

14.6 Weight Matrix for Differenced Observations.......465

14.7 Geometry of the Ellipsoid........................467

14.8 The Direct and Reverse Problems..................470

14.9 Geodetic Reference System 1980...................471

14.10 Geoid, Ellipsoid, and Datum.....................472

14.11 World Geodetic System 1984......................476

14.12 Coordinate Changes From Datum Changes...........477

15 Processing of GPS Data................................481

15.1 Baseline Computation and M-Files.................481

15.2 Coordinate Changes and Satellite Position........482

15.3 Receiver Position from Pseudoranges..............487

15.4 Separate Ambiguity and Baseline Estimation.......488

15.5 Joint Ambiguity and Baseline Estimation..........494

15.6 The LAMBDA Method for Ambiguities................495

15.7 Sequential Filter for Absolute Position..........499

15.8 Additional Useful Filters........................505

16 Random Processes......................................515

16.1 Random Processes in Continuous Time..............515

16.2 Random Processes in Discrete Time................523

16.3 Modeling.........................................5 27

17 Kalman Filters........................................543

17.1 Updating Least Squares...........................543

17.2 Static and Dynamic Updates.......................548

17.3 The Steady Model.................................552

17.4 Derivation of the Kalman Filter..................558

17.5 Bayes Filter for Batch Processing................566

17.6 Smoothing........................................5 69

17.7 An Example from Practice.........................574

The Receiver Independent Exchange Format.................585

Glossary.......................................... .......601

References........................................ .......609

Index of M-files.........................................615

Index............................................. .......617

(E-Mail Removed) http://www-math.mit.edu/~gs