Lisp is the family of most powerful (simple and expressive) programming languages available to date. Scheme is one of its most elegant dialects. (For a fast overview of Scheme see Appendix: Scheme of SICM; for a more detailed study see SICP; for Lisp motivational essays see here and here.)
Scheme is not a common language. This implies fewer available libraries than mainstream choices, and difficulty in interfacing Scheme programs to large collaborations' code bases. Furthermore, there are different implementations and dialects, leading to a fragmented community. A program written for one implementation is likely not to work with another one. The language standards are themselves controversial, having R6RS introduced significant breaks over previous versions, and over established libraries supported by the community (SRFI).
This is a comparison of four Scheme implementations/dialects (Chez, Chicken, Gambit, Racket) in view of modern scientific computing needs (see this post for a more general guide). These include purely numerical applications, but also symbolic computations. See scmutils (online README) for an exemplar package that shows the expressiveness of Scheme.
For each item the implementations in bold outperform the others. Items are listed roughly from the most to the less important. Technical details refer to information retrieved on October, 2018. Tests where performed under Debian 4.18.10-2 (2018-10-07) x86_64 GNU/Linux.
According to these benchmarks the fastest and most reliable Scheme implementations with larger community are: Chez, Chicken, Gambit, Racket. All are available as Debian packages and also provide simple installation from source.
(Bigloo (R5RS) and Larceny (R5RS, R6RS, R7RS) also perform well, but they are not available as Debian packages.)
That said, the kind of benchmarks mentioned above should be taken with a grain of salt. They only consider runtime efficiency (not always optimally, see the Chicken Scheme case) and neglect development time. Python is one of the most popular languages for science, yet it (CPython) can be easily perform 100x slower than C. Having a Scheme implementation that is only 1x-10x slower than C/C++ is a significant improvement. It is also important to keep in mind that the variance between programmers can be larger than the variance between languages, suggesting that an appropriate programming style is more important for overall efficiency than the particular language choice (provided they are in the same ballpark).
The Foreign Function Interface (to call C programs) is important given the lack of native libraries.
All implementations are also supported by SLIB that has a nice inline documentation language, Schmooz.
That said, all implementations are supported in Emacs.
R6RS is controversial, incompatible with R5RS and with SRFI. It has unnecessary features (like square parenthesis equivalent to round parentheses), but also useful ones (e.g., exceptions, libraries) compared to R5RS.
R7RS-small is the most recent minimalistic Scheme revision, improving over controversial R6RS.
Here we discuss more in details similar implementations.
Gambit is smaller and simpler, and documentation more clear. Chez is larger and its documentation is more extended; however, sometimes unclear, e.g., when discussing software distribution (looks like distribution is only supported via Petite Chez?). Chez has a R6RS-compliant library support, but it has limited 3rd-party libraries availability and no established package manager; Gambit has its own library system and a few 3rd-party libraries are available via semi-official package managers, but often their installation is difficult and links to documentation are broken.
Working only with Gambit would mainly require to write interfaces to C programs even for most basic tasks; distribution as shared libraries is nice, but the process requires cumbersome makefiles. Chez is better supported by 3rd party libraries and The Scheme Programming Language (written by Chez author) targets specifically numerical examples. Among the two, at the present Chez seems better suited for numerical computing.
Chicken has reliable package manager, libraries and an helpful lively community (it is also the only Scheme discussed here whose source files are available on self-hosted git repository). There are extensive tutorials on several practical aspects of the language, also targeted to programmers coming from other languages. The compiler requires experimenting (and can be quite slow with inappropriate options), but it allows large room for optimization if needed. Online publicly available benchmarks put Chicken in the ballpark 10x slower than C, comparable to Racket but with a significantly smaller variance (especially more convenient when capturing continuations).
Racket has many nice features. Racket main language is not RnRS-compatible, not necessarily a (dis-)advantage. It has an excellent package manager and a large library repository. It is maybe the most friendly implementation for people coming from Python. However, embedding Racket into C will be likely challenging if needed. Performance has a large variance, use of continuations and I/O operations can seriously slow down programs by a factor O(10-100). Overall it is placed in the ballpark of 10x slower than C (see also Benchmarks Game).
We compared Chez, Chicken, Gambit and Racket in view of usage in scientific programming. They can be divided into two groups:
Unfortunately the lack of scientific libraries is a serious practical issue and there is no ideal implementation to promptly replace popular languages like Python. Non-negligible efforts are likely required to write from scratch wrappers to C/C++/Fortran libraries. The absence of a package manager in Chez and Gambit makes it too unpractical to use these highly efficient and reliable implementations. Due to Racket large variance in runtime efficiency (especially when capturing continuations) and in packages quality, Chicken seems the most suited candidate for a Scheme implementation usable in scientific computing tasks.
A significant community effort around Chez (starting from establishing a package manager) is much desirable. As Chez is being adopted by larger projects (e.g., there are plans to implement Racket itself in Chez), a critical mass of users may be reached.
An obvious alternative to Scheme as Lisp dialect is Common Lisp. Common Lisp has excellent, efficient (runtime usually of the same order as C, only a few factors slower) battery-included implementations like SBCL (same ballpark, slightly faster than Chez Scheme), a large community, an established package manager to easily access an extensive library (quicklisp) that receives frequent several new contributions, and good support for docstrings.
Scheme is conceptually simpler than Common Lisp thanks to only one namespace for functions and variables (among other differences). E.g., Scheme only needs define for both variables and functions, there are no defun, setf, defvar, etc. (but, of course, they can be defined if needed as an extension of Scheme, e.g., in a Domain Specific Language). Especially, Scheme does not need the awkward funcall for function application.
A detailed comparison of the practical advantages for scientific computing of Scheme, Common Lisp or other Lisp dialects (like Clojure) is worth a separate discussion.